## 5.1 Quadratic Functions

Learning objectives.

In this section, you will:

- Recognize characteristics of parabolas.
- Understand how the graph of a parabola is related to its quadratic function.
- Determine a quadratic functionβs minimum or maximum value.
- Solve problems involving a quadratic functionβs minimum or maximum value.

Curved antennas, such as the ones shown in Figure 1 , are commonly used to focus microwaves and radio waves to transmit television and telephone signals, as well as satellite and spacecraft communication. The cross-section of the antenna is in the shape of a parabola, which can be described by a quadratic function.

In this section, we will investigate quadratic functions, which frequently model problems involving area and projectile motion. Working with quadratic functions can be less complex than working with higher degree functions, so they provide a good opportunity for a detailed study of function behavior.

## Recognizing Characteristics of Parabolas

The graph of a quadratic function is a U-shaped curve called a parabola . One important feature of the graph is that it has an extreme point, called the vertex . If the parabola opens up, the vertex represents the lowest point on the graph, or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph, or the maximum value . In either case, the vertex is a turning point on the graph. The graph is also symmetric with a vertical line drawn through the vertex, called the axis of symmetry . These features are illustrated in Figure 2 .

The y -intercept is the point at which the parabola crosses the y -axis. The x -intercepts are the points at which the parabola crosses the x -axis. If they exist, the x -intercepts represent the zeros , or roots , of the quadratic function, the values of x x at which y = 0. y = 0.

## Identifying the Characteristics of a Parabola

Determine the vertex, axis of symmetry, zeros, and y - y - intercept of the parabola shown in Figure 3 .

The vertex is the turning point of the graph. We can see that the vertex is at ( 3 , 1 ) . ( 3 , 1 ) . Because this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry is x = 3. x = 3. This parabola does not cross the x - x - axis, so it has no zeros. It crosses the y - y - axis at ( 0 , 7 ) ( 0 , 7 ) so this is the y -intercept.

## Understanding How the Graphs of Parabolas are Related to Their Quadratic Functions

The general form of a quadratic function presents the function in the form

where a , b , a , b , and c c are real numbers and a β 0. a β 0. If a > 0 , a > 0 , the parabola opens upward. If a < 0 , a < 0 , the parabola opens downward. We can use the general form of a parabola to find the equation for the axis of symmetry.

The axis of symmetry is defined by x = β b 2 a . x = β b 2 a . If we use the quadratic formula, x = β b Β± b 2 β 4 a c 2 a , x = β b Β± b 2 β 4 a c 2 a , to solve a x 2 + b x + c = 0 a x 2 + b x + c = 0 for the x - x - intercepts, or zeros, we find the value of x x halfway between them is always x = β b 2 a , x = β b 2 a , the equation for the axis of symmetry.

Figure 4 represents the graph of the quadratic function written in general form as y = x 2 + 4 x + 3. y = x 2 + 4 x + 3. In this form, a = 1 , b = 4 , a = 1 , b = 4 , and c = 3. c = 3. Because a > 0 , a > 0 , the parabola opens upward. The axis of symmetry is x = β 4 2 ( 1 ) = β2. x = β 4 2 ( 1 ) = β2. This also makes sense because we can see from the graph that the vertical line x = β2 x = β2 divides the graph in half. The vertex always occurs along the axis of symmetry. For a parabola that opens upward, the vertex occurs at the lowest point on the graph, in this instance, ( β2 , β1 ) . ( β2 , β1 ) . The x - x - intercepts, those points where the parabola crosses the x - x - axis, occur at ( β3 , 0 ) ( β3 , 0 ) and ( β1 , 0 ) . ( β1 , 0 ) .

The standard form of a quadratic function presents the function in the form

where ( h , k ) ( h , k ) is the vertex. Because the vertex appears in the standard form of the quadratic function, this form is also known as the vertex form of a quadratic function .

As with the general form, if a > 0 , a > 0 , the parabola opens upward and the vertex is a minimum. If a < 0 , a < 0 , the parabola opens downward, and the vertex is a maximum. Figure 5 represents the graph of the quadratic function written in standard form as y = β3 ( x + 2 ) 2 + 4. y = β3 ( x + 2 ) 2 + 4. Since x β h = x + 2 x β h = x + 2 in this example, h = β2. h = β2. In this form, a = β3 , h = β2 , a = β3 , h = β2 , and k = 4. k = 4. Because a < 0 , a < 0 , the parabola opens downward. The vertex is at ( β 2 , 4 ) . ( β 2 , 4 ) .

The standard form is useful for determining how the graph is transformed from the graph of y = x 2 . y = x 2 . Figure 6 is the graph of this basic function.

If k > 0 , k > 0 , the graph shifts upward, whereas if k < 0 , k < 0 , the graph shifts downward. In Figure 5 , k > 0 , k > 0 , so the graph is shifted 4 units upward. If h > 0 , h > 0 , the graph shifts toward the right and if h < 0 , h < 0 , the graph shifts to the left. In Figure 5 , h < 0 , h < 0 , so the graph is shifted 2 units to the left. The magnitude of a a indicates the stretch of the graph. If | a | > 1 , | a | > 1 , the point associated with a particular x - x - value shifts farther from the x- axis, so the graph appears to become narrower, and there is a vertical stretch. But if | a | < 1 , | a | < 1 , the point associated with a particular x - x - value shifts closer to the x- axis, so the graph appears to become wider, but in fact there is a vertical compression. In Figure 5 , | a | > 1 , | a | > 1 , so the graph becomes narrower.

The standard form and the general form are equivalent methods of describing the same function. We can see this by expanding out the general form and setting it equal to the standard form.

For the linear terms to be equal, the coefficients must be equal.

This is the axis of symmetry we defined earlier. Setting the constant terms equal:

In practice, though, it is usually easier to remember that k is the output value of the function when the input is h , h , so f ( h ) = k . f ( h ) = k .

## Forms of Quadratic Functions

A quadratic function is a polynomial function of degree two. The graph of a quadratic function is a parabola.

The general form of a quadratic function is f ( x ) = a x 2 + b x + c f ( x ) = a x 2 + b x + c where a , b , a , b , and c c are real numbers and a β 0. a β 0.

The standard form of a quadratic function is f ( x ) = a ( x β h ) 2 + k f ( x ) = a ( x β h ) 2 + k where a β 0. a β 0.

The vertex ( h , k ) ( h , k ) is located at

Given a graph of a quadratic function, write the equation of the function in general form.

- Identify the horizontal shift of the parabola; this value is h . h . Identify the vertical shift of the parabola; this value is k . k .
- Substitute the values of the horizontal and vertical shift for h h and k . k . in the function f ( x ) = a ( x β h ) 2 + k . f ( x ) = a ( x β h ) 2 + k .
- Substitute the values of any point, other than the vertex, on the graph of the parabola for x x and f ( x ) . f ( x ) .
- Solve for the stretch factor, | a | . | a | .
- Expand and simplify to write in general form.

## Writing the Equation of a Quadratic Function from the Graph

Write an equation for the quadratic function g g in Figure 7 as a transformation of f ( x ) = x 2 , f ( x ) = x 2 , and then expand the formula, and simplify terms to write the equation in general form.

We can see the graph of g is the graph of f ( x ) = x 2 f ( x ) = x 2 shifted to the left 2 and down 3, giving a formula in the form g ( x ) = a ( x β ( β2 ) ) 2 β 3 = a ( x + 2 ) 2 β 3. g ( x ) = a ( x β ( β2 ) ) 2 β 3 = a ( x + 2 ) 2 β 3.

Substituting the coordinates of a point on the curve, such as ( 0 , β1 ) , ( 0 , β1 ) , we can solve for the stretch factor.

In standard form, the algebraic model for this graph is ( g ) x = 1 2 ( x + 2 ) 2 β 3. ( g ) x = 1 2 ( x + 2 ) 2 β 3.

To write this in general polynomial form, we can expand the formula and simplify terms.

Notice that the horizontal and vertical shifts of the basic graph of the quadratic function determine the location of the vertex of the parabola; the vertex is unaffected by stretches and compressions.

We can check our work using the table feature on a graphing utility. First enter Y1 = 1 2 ( x + 2 ) 2 β 3. Y1 = 1 2 ( x + 2 ) 2 β 3. Next, select TBLSET, TBLSET, then use TblStart = β 6 TblStart = β 6 and Ξ Tbl = 2, Ξ Tbl = 2, and select TABLE . TABLE . See Table 1 .

β6 | β4 | β2 | 0 | 2 | |

5 | β1 | β3 | β1 | 5 |

The ordered pairs in the table correspond to points on the graph.

A coordinate grid has been superimposed over the quadratic path of a basketball in Figure 8 . Assume that the point (β4, 7) is the highest point of the basketballβs trajectory. Find an equation for the path of the ball. Does the shooter make the basket?

Given a quadratic function in general form, find the vertex of the parabola.

- Identify a , b , and c . a , b , and c .
- Find h , h , the x -coordinate of the vertex, by substituting a a and b b into h = β b 2 a . h = β b 2 a .
- Find k , k , the y -coordinate of the vertex, by evaluating k = f ( h ) = f ( β b 2 a ) . k = f ( h ) = f ( β b 2 a ) .

## Finding the Vertex of a Quadratic Function

Find the vertex of the quadratic function f ( x ) = 2 x 2 β 6 x + 7. f ( x ) = 2 x 2 β 6 x + 7. Rewrite the quadratic in standard form (vertex form).

The horizontal coordinate of the vertex will be at h = β b 2 a = β β6 2 ( 2 ) = 6 4 = 3 2 The vertical coordinate of the vertex will be at k = f ( h ) = f ( 3 2 ) = 2 ( 3 2 ) 2 β 6 ( 3 2 ) + 7 = 5 2 The horizontal coordinate of the vertex will be at h = β b 2 a = β β6 2 ( 2 ) = 6 4 = 3 2 The vertical coordinate of the vertex will be at k = f ( h ) = f ( 3 2 ) = 2 ( 3 2 ) 2 β 6 ( 3 2 ) + 7 = 5 2

Rewriting into standard form, the stretch factor will be the same as the a a in the original quadratic. First, find the horizontal coordinate of the vertex. Then find the vertical coordinate of the vertex. Substitute the values into standard form, using the " a a " from the general form.

The standard form of a quadratic function prior to writing the function then becomes the following:

One reason we may want to identify the vertex of the parabola is that this point will inform us where the maximum or minimum value of the output occurs, k , k , and where it occurs, x . x .

Given the equation g ( x ) = 13 + x 2 β 6 x , g ( x ) = 13 + x 2 β 6 x , write the equation in general form and then in standard form.

## Finding the Domain and Range of a Quadratic Function

Any number can be the input value of a quadratic function. Therefore, the domain of any quadratic function is all real numbers. Because parabolas have a maximum or a minimum point, the range is restricted. Since the vertex of a parabola will be either a maximum or a minimum, the range will consist of all y -values greater than or equal to the y -coordinate at the turning point or less than or equal to the y -coordinate at the turning point, depending on whether the parabola opens up or down.

## Domain and Range of a Quadratic Function

The domain of any quadratic function is all real numbers unless the context of the function presents some restrictions.

The range of a quadratic function written in general form f ( x ) = a x 2 + b x + c f ( x ) = a x 2 + b x + c with a positive a a value is f ( x ) β₯ f ( β b 2 a ) , f ( x ) β₯ f ( β b 2 a ) , or [ f ( β b 2 a ) , β ) ; [ f ( β b 2 a ) , β ) ; the range of a quadratic function written in general form with a negative a a value is f ( x ) β€ f ( β b 2 a ) , f ( x ) β€ f ( β b 2 a ) , or ( β β , f ( β b 2 a ) ] . ( β β , f ( β b 2 a ) ] .

The range of a quadratic function written in standard form f ( x ) = a ( x β h ) 2 + k f ( x ) = a ( x β h ) 2 + k with a positive a a value is f ( x ) β₯ k ; f ( x ) β₯ k ; the range of a quadratic function written in standard form with a negative a a value is f ( x ) β€ k . f ( x ) β€ k .

Given a quadratic function, find the domain and range.

- Identify the domain of any quadratic function as all real numbers.
- Determine whether a a is positive or negative. If a a is positive, the parabola has a minimum. If a a is negative, the parabola has a maximum.
- Determine the maximum or minimum value of the parabola, k . k .
- If the parabola has a minimum, the range is given by f ( x ) β₯ k , f ( x ) β₯ k , or [ k , β ) . [ k , β ) . If the parabola has a maximum, the range is given by f ( x ) β€ k , f ( x ) β€ k , or ( β β , k ] . ( β β , k ] .

Find the domain and range of f ( x ) = β 5 x 2 + 9 x β 1. f ( x ) = β 5 x 2 + 9 x β 1.

As with any quadratic function, the domain is all real numbers.

Because a a is negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding the x - x - value of the vertex.

The maximum value is given by f ( h ) . f ( h ) .

The range is f ( x ) β€ 61 20 , f ( x ) β€ 61 20 , or ( β β , 61 20 ] . ( β β , 61 20 ] .

Find the domain and range of f ( x ) = 2 ( x β 4 7 ) 2 + 8 11 . f ( x ) = 2 ( x β 4 7 ) 2 + 8 11 .

## Determining the Maximum and Minimum Values of Quadratic Functions

The output of the quadratic function at the vertex is the maximum or minimum value of the function, depending on the orientation of the parabola . We can see the maximum and minimum values in Figure 9 .

There are many real-world scenarios that involve finding the maximum or minimum value of a quadratic function, such as applications involving area and revenue.

## Finding the Maximum Value of a Quadratic Function

A backyard farmer wants to enclose a rectangular space for a new garden within her fenced backyard. She has purchased 80 feet of wire fencing to enclose three sides, and she will use a section of the backyard fence as the fourth side.

- β Find a formula for the area enclosed by the fence if the sides of fencing perpendicular to the existing fence have length L . L .
- β What dimensions should she make her garden to maximize the enclosed area?

Letβs use a diagram such as Figure 10 to record the given information. It is also helpful to introduce a temporary variable, W , W , to represent the width of the garden and the length of the fence section parallel to the backyard fence.

Now we are ready to write an equation for the area the fence encloses. We know the area of a rectangle is length multiplied by width, so

This formula represents the area of the fence in terms of the variable length L . L . The function, written in general form, is

- β The quadratic has a negative leading coefficient, so the graph will open downward, and the vertex will be the maximum value for the area. In finding the vertex, we must be careful because the equation is not written in standard polynomial form with decreasing powers. This is why we rewrote the function in general form above. Since a a is the coefficient of the squared term, a = β2 , b = 80 , a = β2 , b = 80 , and c = 0. c = 0.

To find the vertex:

The maximum value of the function is an area of 800 square feet, which occurs when L = 20 L = 20 feet. When the shorter sides are 20 feet, there is 40 feet of fencing left for the longer side. To maximize the area, she should enclose the garden so the two shorter sides have length 20 feet and the longer side parallel to the existing fence has length 40 feet.

This problem also could be solved by graphing the quadratic function. We can see where the maximum area occurs on a graph of the quadratic function in Figure 11 .

Given an application involving revenue, use a quadratic equation to find the maximum.

- Write a quadratic equation for a revenue function.
- Find the vertex of the quadratic equation.
- Determine the y -value of the vertex.

## Finding Maximum Revenue

The unit price of an item affects its supply and demand. That is, if the unit price goes up, the demand for the item will usually decrease. For example, a local newspaper currently has 84,000 subscribers at a quarterly charge of $30. Market research has suggested that if the owners raise the price to $32, they would lose 5,000 subscribers. Assuming that subscriptions are linearly related to the price, what price should the newspaper charge for a quarterly subscription to maximize their revenue?

Revenue is the amount of money a company brings in. In this case, the revenue can be found by multiplying the price per subscription times the number of subscribers, or quantity. We can introduce variables, p p for price per subscription and Q Q for quantity, giving us the equation Revenue = p Q . Revenue = p Q .

Because the number of subscribers changes with the price, we need to find a relationship between the variables. We know that currently p = 30 p = 30 and Q = 84,000. Q = 84,000. We also know that if the price rises to $32, the newspaper would lose 5,000 subscribers, giving a second pair of values, p = 32 p = 32 and Q = 79,000. Q = 79,000. From this we can find a linear equation relating the two quantities. The slope will be

This tells us the paper will lose 2,500 subscribers for each dollar they raise the price. We can then solve for the y -intercept.

This gives us the linear equation Q = β2,500 p + 159,000 Q = β2,500 p + 159,000 relating cost and subscribers. We now return to our revenue equation.

We now have a quadratic function for revenue as a function of the subscription charge. To find the price that will maximize revenue for the newspaper, we can find the vertex.

The model tells us that the maximum revenue will occur if the newspaper charges $31.80 for a subscription. To find what the maximum revenue is, we evaluate the revenue function.

This could also be solved by graphing the quadratic as in Figure 12 . We can see the maximum revenue on a graph of the quadratic function.

## Finding the x - and y -Intercepts of a Quadratic Function

Much as we did in the application problems above, we also need to find intercepts of quadratic equations for graphing parabolas. Recall that we find the y - y - intercept of a quadratic by evaluating the function at an input of zero, and we find the x - x - intercepts at locations where the output is zero. Notice in Figure 13 that the number of x - x - intercepts can vary depending upon the location of the graph.

Given a quadratic function f ( x ) , f ( x ) , find the y - y - and x -intercepts.

- Evaluate f ( 0 ) f ( 0 ) to find the y -intercept.
- Solve the quadratic equation f ( x ) = 0 f ( x ) = 0 to find the x -intercepts.

## Finding the y - and x -Intercepts of a Parabola

Find the y - and x -intercepts of the quadratic f ( x ) = 3 x 2 + 5 x β 2. f ( x ) = 3 x 2 + 5 x β 2.

We find the y -intercept by evaluating f ( 0 ) . f ( 0 ) .

So the y -intercept is at ( 0 , β2 ) . ( 0 , β2 ) .

For the x -intercepts, we find all solutions of f ( x ) = 0. f ( x ) = 0.

In this case, the quadratic can be factored easily, providing the simplest method for solution.

So the x -intercepts are at ( 1 3 , 0 ) ( 1 3 , 0 ) and ( β 2 , 0 ) . ( β 2 , 0 ) .

By graphing the function, we can confirm that the graph crosses the y -axis at ( 0 , β2 ) . ( 0 , β2 ) . We can also confirm that the graph crosses the x -axis at ( 1 3 , 0 ) ( 1 3 , 0 ) and ( β2 , 0 ) . ( β2 , 0 ) . See Figure 14

## Rewriting Quadratics in Standard Form

In Example 7 , the quadratic was easily solved by factoring. However, there are many quadratics that cannot be factored. We can solve these quadratics by first rewriting them in standard form.

Given a quadratic function, find the x - x - intercepts by rewriting in standard form .

- Substitute a a and b b into h = β b 2 a . h = β b 2 a .
- Substitute x = h x = h into the general form of the quadratic function to find k . k .
- Rewrite the quadratic in standard form using h h and k . k .
- Solve for when the output of the function will be zero to find the x - x - intercepts.

## Finding the x -Intercepts of a Parabola

Find the x - x - intercepts of the quadratic function f ( x ) = 2 x 2 + 4 x β 4. f ( x ) = 2 x 2 + 4 x β 4.

We begin by solving for when the output will be zero.

Because the quadratic is not easily factorable in this case, we solve for the intercepts by first rewriting the quadratic in standard form.

We know that a = 2. a = 2. Then we solve for h h and k . k .

So now we can rewrite in standard form.

We can now solve for when the output will be zero.

The graph has x -intercepts at ( β1 β 3 , 0 ) ( β1 β 3 , 0 ) and ( β1 + 3 , 0 ) . ( β1 + 3 , 0 ) .

We can check our work by graphing the given function on a graphing utility and observing the x - x - intercepts. See Figure 15 .

We could have achieved the same results using the quadratic formula. Identify a = 2 , b = 4 a = 2 , b = 4 and c = β4. c = β4.

So the x -intercepts occur at ( β 1 β 3 , 0 ) ( β 1 β 3 , 0 ) and ( β 1 + 3 , 0 ) . ( β 1 + 3 , 0 ) .

In a Try It , we found the standard and general form for the function g ( x ) = 13 + x 2 β 6 x . g ( x ) = 13 + x 2 β 6 x . Now find the y - and x -intercepts (if any).

## Applying the Vertex and x -Intercepts of a Parabola

A ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second. The ballβs height above ground can be modeled by the equation H ( t ) = β 16 t 2 + 80 t + 40. H ( t ) = β 16 t 2 + 80 t + 40.

- β When does the ball reach the maximum height?
- β What is the maximum height of the ball?
- β When does the ball hit the ground?

The ball reaches a maximum height after 2.5 seconds.

The ball reaches a maximum height of 140 feet.

We use the quadratic formula.

Because the square root does not simplify nicely, we can use a calculator to approximate the values of the solutions.

The second answer is outside the reasonable domain of our model, so we conclude the ball will hit the ground after about 5.458 seconds. See Figure 16 .

Note that the graph does not represent the physical path of the ball upward and downward. Keep the quantities on each axis in mind while interpreting the graph.

A rock is thrown upward from the top of a 112-foot high cliff overlooking the ocean at a speed of 96 feet per second. The rockβs height above ocean can be modeled by the equation H ( t ) = β16 t 2 + 96 t + 112. H ( t ) = β16 t 2 + 96 t + 112.

- β When does the rock reach the maximum height?
- β What is the maximum height of the rock?
- β When does the rock hit the ocean?

Access these online resources for additional instruction and practice with quadratic equations.

- Graphing Quadratic Functions in General Form
- Graphing Quadratic Functions in Standard Form
- Quadratic Function Review
- Characteristics of a Quadratic Function

## 5.1 Section Exercises

Explain the advantage of writing a quadratic function in standard form.

How can the vertex of a parabola be used in solving real-world problems?

Explain why the condition of a β 0 a β 0 is imposed in the definition of the quadratic function.

What is another name for the standard form of a quadratic function?

What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?

For the following exercises, rewrite the quadratic functions in standard form and give the vertex.

f ( x ) = x 2 β 12 x + 32 f ( x ) = x 2 β 12 x + 32

g ( x ) = x 2 + 2 x β 3 g ( x ) = x 2 + 2 x β 3

f ( x ) = x 2 β x f ( x ) = x 2 β x

f ( x ) = x 2 + 5 x β 2 f ( x ) = x 2 + 5 x β 2

h ( x ) = 2 x 2 + 8 x β 10 h ( x ) = 2 x 2 + 8 x β 10

k ( x ) = 3 x 2 β 6 x β 9 k ( x ) = 3 x 2 β 6 x β 9

f ( x ) = 2 x 2 β 6 x f ( x ) = 2 x 2 β 6 x

f ( x ) = 3 x 2 β 5 x β 1 f ( x ) = 3 x 2 β 5 x β 1

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.

y ( x ) = 2 x 2 + 10 x + 12 y ( x ) = 2 x 2 + 10 x + 12

f ( x ) = 2 x 2 β 10 x + 4 f ( x ) = 2 x 2 β 10 x + 4

f ( x ) = β x 2 + 4 x + 3 f ( x ) = β x 2 + 4 x + 3

f ( x ) = 4 x 2 + x β 1 f ( x ) = 4 x 2 + x β 1

h ( t ) = β4 t 2 + 6 t β 1 h ( t ) = β4 t 2 + 6 t β 1

f ( x ) = 1 2 x 2 + 3 x + 1 f ( x ) = 1 2 x 2 + 3 x + 1

f ( x ) = β 1 3 x 2 β 2 x + 3 f ( x ) = β 1 3 x 2 β 2 x + 3

For the following exercises, determine the domain and range of the quadratic function.

f ( x ) = ( x β 3 ) 2 + 2 f ( x ) = ( x β 3 ) 2 + 2

f ( x ) = β2 ( x + 3 ) 2 β 6 f ( x ) = β2 ( x + 3 ) 2 β 6

f ( x ) = x 2 + 6 x + 4 f ( x ) = x 2 + 6 x + 4

f ( x ) = 2 x 2 β 4 x + 2 f ( x ) = 2 x 2 β 4 x + 2

For the following exercises, use the vertex ( h , k ) ( h , k ) and a point on the graph ( x , y ) ( x , y ) to find the general form of the equation of the quadratic function.

( h , k ) = ( 2 , 0 ) , ( x , y ) = ( 4 , 4 ) ( h , k ) = ( 2 , 0 ) , ( x , y ) = ( 4 , 4 )

( h , k ) = ( β2 , β1 ) , ( x , y ) = ( β4 , 3 ) ( h , k ) = ( β2 , β1 ) , ( x , y ) = ( β4 , 3 )

( h , k ) = ( 0 , 1 ) , ( x , y ) = ( 2 , 5 ) ( h , k ) = ( 0 , 1 ) , ( x , y ) = ( 2 , 5 )

( h , k ) = ( 2 , 3 ) , ( x , y ) = ( 5 , 12 ) ( h , k ) = ( 2 , 3 ) , ( x , y ) = ( 5 , 12 )

( h , k ) = ( β 5 , 3 ) , ( x , y ) = ( 2 , 9 ) ( h , k ) = ( β 5 , 3 ) , ( x , y ) = ( 2 , 9 )

( h , k ) = ( 3 , 2 ) , ( x , y ) = ( 10 , 1 ) ( h , k ) = ( 3 , 2 ) , ( x , y ) = ( 10 , 1 )

( h , k ) = ( 0 , 1 ) , ( x , y ) = ( 1 , 0 ) ( h , k ) = ( 0 , 1 ) , ( x , y ) = ( 1 , 0 )

( h , k ) = ( 1 , 0 ) , ( x , y ) = ( 0 , 1 ) ( h , k ) = ( 1 , 0 ) , ( x , y ) = ( 0 , 1 )

For the following exercises, sketch a graph of the quadratic function and give the vertex, axis of symmetry, and intercepts.

f ( x ) = x 2 β 2 x f ( x ) = x 2 β 2 x

f ( x ) = x 2 β 6 x β 1 f ( x ) = x 2 β 6 x β 1

f ( x ) = x 2 β 5 x β 6 f ( x ) = x 2 β 5 x β 6

f ( x ) = x 2 β 7 x + 3 f ( x ) = x 2 β 7 x + 3

f ( x ) = β2 x 2 + 5 x β 8 f ( x ) = β2 x 2 + 5 x β 8

f ( x ) = 4 x 2 β 12 x β 3 f ( x ) = 4 x 2 β 12 x β 3

For the following exercises, write the equation for the graphed quadratic function.

For the following exercises, use the table of values that represent points on the graph of a quadratic function. By determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function.

β2 | β1 | 0 | 1 | 2 | |

5 | 2 | 1 | 2 | 5 |

β2 | β1 | 0 | 1 | 2 | |

1 | 0 | 1 | 4 | 9 |

β2 | β1 | 0 | 1 | 2 | |

β2 | 1 | 2 | 1 | β2 |

β2 | β1 | 0 | 1 | 2 | |

β8 | β3 | 0 | 1 | 0 |

β2 | β1 | 0 | 1 | 2 | |

8 | 2 | 0 | 2 | 8 |

For the following exercises, use a calculator to find the answer.

Graph on the same set of axes the functions f ( x ) = x 2 f ( x ) = x 2 , f ( x ) = 2 x 2 f ( x ) = 2 x 2 , and f ( x ) = 1 3 x 2 f ( x ) = 1 3 x 2 .

What appears to be the effect of changing the coefficient?

Graph on the same set of axes f ( x ) = x 2 , f ( x ) = x 2 + 2 f ( x ) = x 2 , f ( x ) = x 2 + 2 and f ( x ) = x 2 , f ( x ) = x 2 + 5 f ( x ) = x 2 , f ( x ) = x 2 + 5 and f ( x ) = x 2 β 3. f ( x ) = x 2 β 3. What appears to be the effect of adding a constant?

Graph on the same set of axes f ( x ) = x 2 , f ( x ) = ( x β 2 ) 2 , f ( x β 3 ) 2 f ( x ) = x 2 , f ( x ) = ( x β 2 ) 2 , f ( x β 3 ) 2 , and f ( x ) = ( x + 4 ) 2 . f ( x ) = ( x + 4 ) 2 .

What appears to be the effect of adding or subtracting those numbers?

The path of an object projected at a 45 degree angle with initial velocity of 80 feet per second is given by the function h ( x ) = β 32 ( 80 ) 2 x 2 + x h ( x ) = β 32 ( 80 ) 2 x 2 + x where x x is the horizontal distance traveled and h ( x ) h ( x ) is the height in feet. Use the TRACE feature of your calculator to determine the height of the object when it has traveled 100 feet away horizontally.

A suspension bridge can be modeled by the quadratic function h ( x ) = .0001 x 2 h ( x ) = .0001 x 2 with β2000 β€ x β€ 2000 β2000 β€ x β€ 2000 where | x | | x | is the number of feet from the center and h ( x ) h ( x ) is height in feet. Use the TRACE feature of your calculator to estimate how far from the center does the bridge have a height of 100 feet.

For the following exercises, use the vertex of the graph of the quadratic function and the direction the graph opens to find the domain and range of the function.

Vertex ( 1 , β2 ) , ( 1 , β2 ) , opens up.

Vertex ( β1 , 2 ) ( β1 , 2 ) opens down.

Vertex ( β5 , 11 ) , ( β5 , 11 ) , opens down.

Vertex ( β100 , 100 ) , ( β100 , 100 ) , opens up.

For the following exercises, write the equation of the quadratic function that contains the given point and has the same shape as the given function.

Contains ( 1 , 1 ) ( 1 , 1 ) and has shape of f ( x ) = 2 x 2 . f ( x ) = 2 x 2 . Vertex is on the y - y - axis.

Contains ( β1 , 4 ) ( β1 , 4 ) and has the shape of f ( x ) = 2 x 2 . f ( x ) = 2 x 2 . Vertex is on the y - y - axis.

Contains ( 2 , 3 ) ( 2 , 3 ) and has the shape of f ( x ) = 3 x 2 . f ( x ) = 3 x 2 . Vertex is on the y - y - axis.

Contains ( 1 , β3 ) ( 1 , β3 ) and has the shape of f ( x ) = β x 2 . f ( x ) = β x 2 . Vertex is on the y - y - axis.

Contains ( 4 , 3 ) ( 4 , 3 ) and has the shape of f ( x ) = 5 x 2 . f ( x ) = 5 x 2 . Vertex is on the y - y - axis.

Contains ( 1 , β6 ) ( 1 , β6 ) has the shape of f ( x ) = 3 x 2 . f ( x ) = 3 x 2 . Vertex has x-coordinate of β1. β1.

## Real-World Applications

Find the dimensions of the rectangular dog park producing the greatest enclosed area given 200 feet of fencing.

Find the dimensions of the rectangular dog park split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing.

Find the dimensions of the rectangular dog park producing the greatest enclosed area split into 3 sections of the same size given 500 feet of fencing.

Among all of the pairs of numbers whose sum is 6, find the pair with the largest product. What is the product?

Among all of the pairs of numbers whose difference is 12, find the pair with the smallest product. What is the product?

Suppose that the price per unit in dollars of a cell phone production is modeled by p = $ 45 β 0.0125 x , p = $ 45 β 0.0125 x , where x x is in thousands of phones produced, and the revenue represented by thousands of dollars is R = x β p . R = x β p . Find the production level that will maximize revenue.

A rocket is launched in the air. Its height, in meters above sea level, as a function of time, in seconds, is given by h ( t ) = β4.9 t 2 + 229 t + 234. h ( t ) = β4.9 t 2 + 229 t + 234. Find the maximum height the rocket attains.

A ball is thrown in the air from the top of a building. Its height, in meters above ground, as a function of time, in seconds, is given by h ( t ) = β 4.9 t 2 + 24 t + 8. h ( t ) = β 4.9 t 2 + 24 t + 8. How long does it take to reach maximum height?

A soccer stadium holds 62,000 spectators. With a ticket price of $11, the average attendance has been 26,000. When the price dropped to $9, the average attendance rose to 31,000. Assuming that attendance is linearly related to ticket price, what ticket price would maximize revenue?

A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

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- Real functions
- Quadratic Function
- How to write a quadratic function in Factored Form

In this article we explain how to write a quadratic function in its factored form , that is, using zeros to get an equation in product form.

Table of Contents

## What is the factored form?

Given a quadratic function *f(x)=ax^2+bx+c*, we can write it as a product using its zeros and leading coefficient. This is known as the factored form of the quadratic function :

*f(x)=a(x-x_1)(x-x_2)*

*x_1* and *x_2* are the zeros of the function.

If the discriminant is zero, the zeros will be equal *x_1=x_2*, and the equation takes the form:

*f(x)=a(x-x_1)^2*

Since we are dealing with real functions , we will not consider the case where the discriminant is negative, as the zeros would be complex numbers.

It is evident that to find the factored form, we must first know the zeros of the function. For this, it is necessary to review this article:

How to find the zeros of a quadratic function

## Solved exercises

Exercise 1: Find the factored form of the function *f(x)=3x^2+3x-18*

Using the quadratic formula, we find the zeros of *f:* *x_1=2, x_2=-3.* Write the function in factored form as:

*f(x)=3(x-2)(x-(-3))*

*f(x)=3(x-2)(x+3)*

Exercise 2: Write the function *f(x)=-5x^2-10x-5* in factored form.

Find the zeros of the function, noticing that there is only one zero with multiplicity two: *x_1=-1.* Write the factored form as:

As *x_1=x_2,* it results in:

*f(x)=-5(x-(-1))^2*

*f(x)=-5(x+1)^2*

Exercise 3: Factorize the quadratic function *f(x)=x^2-16*

Calculate the zeros: *x_1=-4, x_2=4* and write the factored form as:

*f(x)=1(x-(-4))(x-4)*

*f(x)=(x+4)(x-4)*

Exercise 4: Find the factored form of the function *f(x)=-7x^2*

Find the zeros and notice there is only one zero with multiplicity two: *x_1=0,* in this case, it corresponds to the factored form:

*f(x)=-7(x-0)^2*

*f(x)=-7x^2*

We see that in this case, the factored form and the polynomial form coincide.

Daniel Machado

Advanced student of Mathematics at Facultad de Ciencias Exactas, QuΓmicas y Naturales. Universidad Nacional de Misiones, Argentina.

Other articles that may interest you

How to find a quadratic function from its zeros

Find quadratic function from vertex and point

How to write a quadratic function in Vertex Form

Analyzing quadratic functions

Positive and negative intervals of quadratic functions

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## Write the following expression in factored form x ndash 16 x 1 x ndash 16 x ndash 1 x ndash 8 x 2 x ndash 8 x ndash 2

Answered step-by-step

Write the following expression in factored form.

(x – 16)(x + 1)

(x – 16)(x – 1)

(x – 8)(x + 2)

(x – 8)(x – 2)

π€ Not the exact question youβre looking for?

## Related Answered Questions

Let f(x) = ndash3x2 + 5x g(x) = 3 ndash 5x and h(x) = 2x ndash x2 Which of these are TRUE Select all that apply Responsesf(g(x)) = ndash75x2 + 65x ndash 12 f ( g ( x )) = ndash75 x 2 + 65 x ndash 12f(h(x)) = ndash3x4 + 12x3 ndash 12x2 + 5x f ( h ( x )) = ndash3 x 4 + 12 x 3 ndash 12 x 2 + 5 xh(f(x)) = 9x4 ndash 30x3 + 19x2 + 10x h ( f ( x )) = 9 x 4 ndash 30 x 3 + 19 x 2 + 10 xg(g(x)) = 25x ndash 12 g ( g ( x )) = 25 x ndash 12g(h(x)) = 5x2 ndash 10x + 3

What are the zeros of the quadratic function f(x) = 6x2 ndash 24x + 1 x = ndash2 + or x = ndash2 ndash x = 2 + or x = 2 ndash x = ndash2 + or x = ndash2 ndash x = 2 + or x = 2 ndash

Which equation shows how nbsp( ndash10 8) can be used to write the equation of this line in point-slope form y ndash 8 = ndash015(x ndash 10)y + 8 = ndash015(x ndash 10)y ndash 8 = ndash02(x + 10)y + 8 = ndash02(x ndash 10)

Let a and b be real numbers where a nbspb nbsp0 Which of the following functions could represent the graph below nbsp f(x) = x(x ndash a)3 (x ndash b)3 nbspf(x) = (x ndash a)2 (x ndash b)4 nbsp nbspf(x) = x(x ndash a)6 (x ndash b)2 nbsp nbsp f(x) = (x ndash a)5 (x ndash b)

Which of the following accurately lists all discontinuities of the function below point discontinuity at x = ndash2 point discontinuity at x = 4 jump discontinuity at x = ndash2 point discontinuities at x = ndash4 and x = 4 jump discontinuity at x = ndash2 jump discontinuities at x = ndash4 x = ndash2 and x = 4

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## Graphing Quadratics

6th - 9th grade, mathematics.

25 questions

## Introducing new Β Paper mode

No student devices needed. Β Know more

The standard form of a quadratic function is

ax 2 + bx + c

a(x - h) 2 + k

y - y 1 = m(x - x 1 )

Ax + By = C

What do we call the graph of a quadratic?

Identify the 'a' value: y = 16x 2 - 8x -24

Identify the 'b' value: y = 16x 2 - 8x - 24

Identify A, B, and C if y = 3x 2 - 8 + 5x

A=3, B=5, C=-8

A=3, B=-8, C=5

A=3x 2 , B=5x, C=-8

A=3x 2 , B=-8, C=5x

How does the parabola open? y = 4x - x 2 - 1

Which answer choice describes y = -3x 2 + 7x - 2 accurately?

opens up with a maximum

opens up with a minimum

opens down with a maximum

opens down with a minimum

What is the axis of symmetry?

the slope of the graph

the dividing line for a parabola

What is the formula used for?

Finding the axis of symmetry from standard form

Finding the vertex from standard form

Finding the vertex from vertex form

Where is the AoS (Axis of Symmetry) for y = -2x 2 + 4x + 2

Where is the AoS (Axis of Symmetry) for y = 4x 2 + 4

- 13. Multiple Choice Edit 30 seconds 1 pt What is a vertex? where the graph crosses the x-axis the y-intercept the maximum or minimum of the graph where the slope is vertical

What is the green dot on the parabola called?

Find the vertex of y = x 2 - 8x + 14

Find the vertex of y = -2x 2 + 4x + 3

In order to graph a quadratic equation in standard form, the first two steps are to use the equation _____ to find the axis of symmetry, and then you can plug that value back into the equation to find the _____.

-b/2a, vertex

(x-h) 2 , y-intercept

x = 0, solution

ax 2 , vertex

What is the equation for this parabola?

y = x 2 + 2x - 3

y = -x 2 - 4x + 3

y = x 2 + 2x - 4

y = -x 2 + 2x + 3

y = -2/3x 2 - 4x + 7

y = 2/3x 2 - 4x + 7

y = 2/3x 2 - 4x - 7

y = -2/3x 2 - 4x - 7

What are the green dots called?

axis of symmetry

roots, x-intercepts, zeros, or solutions

How many roots does this parabola have?

How do you find the y-intercept?

plug x = 0 into the equation

plug y = 0 into the equation

plug x = 1 into the equation

How do you find the x-intercept?

graphing, square root method, factoring, or quadratic formula

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## Course: Algebra 1 Β > Β Unit 14

The quadratic formula, understanding the quadratic formula.

- Worked example: quadratic formula (example 2)
- Worked example: quadratic formula (negative coefficients)
- Quadratic formula
- Using the quadratic formula: number of solutions
- Number of solutions of quadratic equations
- Quadratic formula review
- Discriminant review

## Worked example

- a β is the coefficient in front of x 2 β , so here a = 1 β (note that a β canβt equal 0 β -- the x 2 β is what makes it a quadratic).
- b β is the coefficient in front of the x β , so here b = 4 β .
- c β is the constant, or the term without any x β next to it, so here c = β 21 β .

## What does the solution tell us?

Second worked example, tips when using the quadratic formula.

- Be careful that the equation is arranged in the right form: a x 2 + b x + c = 0 β or it wonβt work!
- Make sure you take the square root of the whole ( b 2 β 4 a c ) β , and that 2 a β is the denominator of everything above it
- Watch your negatives: b 2 β canβt be negative, so if b β starts as negative, make sure it changes to a positive since the square of a negative or a positive is a positive
- Keep the + / β β and always be on the look out for TWO solutions
- If you use a calculator, the answer might be rounded to a certain number of decimal places. If asked for the exact answer (as usually happens) and the square roots canβt be easily simplified, keep the square roots in the answer, e.g. 2 β 10 2 β and 2 + 10 2 β
- Practice using the quadratic formula .
- Watch Sal do an example:
- Prove the quadratic formula:

## Want to join the conversation?

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## IMAGES

## VIDEO

## COMMENTS

Learn how to graph and analyze quadratic functions in factored form with Quizlet flashcards. Practice finding x-intercepts, vertex, axis of symmetry, domain and range of different quadratic functions. Test your knowledge with interactive questions and feedback.

Explain how you could write a quadratic function in factored form that would have a vertex with an x-coordinate of 3 and two distinct roots. The vertex is on the axis of symmetry, so the axis of symmetry is x = 3.

Study with Quizlet and memorize flashcards containing terms like Which point is an x-intercept of the quadratic function f(x) = (x - 4)(x + 2)?, The graph of the function f(x) = (x + 6)(x + 2) is shown. Which statements describe the graph? Check all that apply., The graph of the function f(x) = -(x + 1)2 is shown. Use the drop-down menus to describe the key aspects of the function. and more.

In the standard form of quadratic equations, there are three parts to it: ax^2 + bx + c where a is the coefficient of the quadratic term, b is the coefficient of the linear term, and c is the constant. The -4 at the end of the equation is the constant. This hopefully answers your last question. Now, your first question.

Quiz 1. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

Vertex FormExample: (x-1)^2+5How to find the vertex:1. Look at the part being squared, so in this case it is (x-1). 2.Find the constant term in the part that is being squared. In this case, the constant is -1. 3. Find the opposite of the constant. In this case the opposite of the constant (-1) is equal to 1.

88 Lesson 3.3 ~ Quadratic Functions in Factored Form step 6: Use what you learned in steps 1-5 to PREDICT what the following graphs will look like. Use your calculator to check your answers. a. y = (x + 9)(x + 2) b.y = 2(x + 3)(x β 1) c. y = βx(x β 6) The x-intercepts of a quadratic function are also called the zeros or roots of the quadratic function.

_____ quadratic function . B. in a parabola, the point at which the function goes from increasing to decreasing or vice versa _____ vertex . C. a point on a line segment that is equidistant from the two endpoints _____ x-intercept . D. a second-degree polynomial function that can be written in the form ax 2 + bx + c, where a, b, and c are real ...

The general form of a quadratic function presents the function in the form. f (x) = a x 2 + b x + c f (x) = a x 2 + b x + c. where a, b, a, b, and c c are real numbers and a ... In this case, the quadratic can be factored easily, providing the simplest method for solution. 0 = (3 x ...

Study with Quizlet and memorize flashcards containing terms like Which is the graph of f(x) = -(x + 3)(x + 1)?, Which function has two x-intercepts, one at (0, 0) and one at (4, 0)?, The graph of the function f(x) = (x - 4)(x + 1) is shown below. ... Quadratic Functions: Factored Form [quiz] 10 terms. Tyonna0520. Preview. AP Calculus Unit 3. 27 ...

Given a quadratic function *f (x)=ax^2+bx+c*, we can write it as a product using its zeros and leading coefficient. This is known as the factored form of the quadratic function: *f (x)=a (x-x_1) (x-x_2)*. *x_1* and *x_2* are the zeros of the function. If the discriminant is zero, the zeros will be equal *x_1=x_2*, and the equation takes the ...

M4 ALGEBRA I. NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 9. Lesson 9: Graphing Quadratic Functions from Factored Form, ππ(ππ )= ππ(ππβππ)(ππβππ. Student Outcomes. Students use the factored form of a quadratic equation to construct a rough graph, use the graph of a quadratic equation to construct ...

The given expression is already in factored form. Each term is a product of two binomials, which is the standard factored form for quadratic expressions. Here's a breakdown of each term: #### Solution By Steps ***Step 1: Identify the factored form of each term*** - The first term is `(x - 16)(x + 1)`. - The second term is `(x - 16)(x - 1)`.

The 1/2 will not change the result created by either factor. Let's say you keep the 1/2 and use it with (x-6) We use the zero product rule: 1/2 (x - 6) = 0. Distribute: x/2 - 3 = 0. Add 3: x/2 = 3. Multiply by 2: x = 6. Notice: this is the same result you would get if you just started with x-6 = 0. Hope this helps.

The graph of the function f(x) = -(x + 1)2 is shown. Use the drop-down menus to describe the key aspects of the function. The vertex is the. The function is positive. The function is decreasing. The domain of the function is. The range of the function is.

True to Form: Forms of Quadratic Functions Lesson Overview Students match quadratic equations with their graphs using key characteristics. The standard form, the factored form, and the vertex form of a quadratic equation are reviewed as is the concavity of a parabola. Students then sort each of the equations with their graphs according to

The standard form of a quadratic function is. ax 2 + bx + c. a(x - h) 2 + k. y - y 1 = m(x - x 1) Ax + By = C. 2. Multiple Choice. Edit. ... In order to graph a quadratic equation in standard form, the first two steps are to use the equation _____ to find the axis of symmetry, and then you can plug that value back into the equation to find the ...

Solve by completing the square: Non-integer solutions. Worked example: completing the square (leading coefficient β 1) Solving quadratics by completing the square: no solution. Proof of the quadratic formula. Solving quadratics by completing the square. Completing the square review. Quadratic formula proof review.

Quizlet has study tools to help you learn anything. Improve your grades and reach your goals with flashcards, practice tests and expert-written solutions today. ... 1 / 10 Quadratic Functions: Factored Form Assignment. Log in. Sign up. Which point is an x-intercept of the quadratic function f(x) = (x - 4)(x + 2)? (-4, 0) (-2, 0) (0, 2) (4, -2 ...

form of a quadratic function. The Standard Form of a Quadratic Equation looks like this: Quadratic Equation There are usually 2 solutions (as shown in the graph above). They are. We want to learn what the shape of a graph of a quadratic function looks like. ANSWER: A Ican r~cognizethe standard form of a quadratic function and

Quiz 1. Learn for free about math, art, computer programming, economics, physics, chemistry, biology, medicine, finance, history, and more. Khan Academy is a nonprofit with the mission of providing a free, world-class education for anyone, anywhere.

MPM2D Quiz 4 *Question 1:* Which of the following is the standard form of a quadratic equation? a) \( y = mx + b \) b) \( ax^2 + bx + c = AI Homework Help. Expert Help. Study Resources. ... What is the y-intercept of the quadratic function \( y = 2x^2 - 3x + 4 \)? a) ... assignment. In relation to completed engagements these are procedures ...

First we need to identify the values for a, b, and c (the coefficients). First step, make sure the equation is in the format from above, a x 2 + b x + c = 0 : x 2 + 4 x β 21 = 0. is what makes it a quadratic). Then we plug a , b , and c into the formula: x = β 4 Β± 16 β 4 β 1 β ( β 21) 2. solving this looks like: