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Polynomial and Rational 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 function’s minimum or maximum value.
• Solve problems involving a quadratic function’s minimum or maximum value.

Curved antennas, such as the ones shown in (Figure), 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).

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 ofat which

### Identifying the Characteristics of a Parabola

Determine the vertex, axis of symmetry, zeros, andintercept of the parabola shown in (Figure).

The vertex is the turning point of the graph. We can see that the vertex is atBecause this parabola opens upward, the axis of symmetry is the vertical line that intersects the parabola at the vertex. So the axis of symmetry isThis parabola does not cross theaxis, so it has no zeros. It crosses theaxis atso 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

whereandare real numbers andIfthe parabola opens upward. Ifthe 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 byIf we use the quadratic formula,to solvefor theintercepts, or zeros, we find the value ofhalfway between them is alwaysthe equation for the axis of symmetry.

(Figure) represents the graph of the quadratic function written in general form asIn this form,andBecausethe parabola opens upward. The axis of symmetry isThis also makes sense because we can see from the graph that the vertical linedivides 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,Theintercepts, those points where the parabola crosses theaxis, occur atand

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

whereis 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, ifthe parabola opens upward and the vertex is a minimum. Ifthe parabola opens downward, and the vertex is a maximum. (Figure) represents the graph of the quadratic function written in standard form asSincein this example,In this form,andBecausethe parabola opens downward. The vertex is at

The standard form is useful for determining how the graph is transformed from the graph of(Figure) is the graph of this basic function.

Ifthe graph shifts upward, whereas ifthe graph shifts downward. In (Figure),so the graph is shifted 4 units upward. Ifthe graph shifts toward the right and ifthe graph shifts to the left. In (Figure),so the graph is shifted 2 units to the left. The magnitude ofindicates the stretch of the graph. If the point associated with a particularvalue shifts farther from the x-axis, so the graph appears to become narrower, and there is a vertical stretch. But ifthe point associated with a particularvalue shifts closer to the x-axis, so the graph appears to become wider, but in fact there is a vertical compression. In (Figure),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 isso

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 iswhereandare real numbers and

The standard form of a quadratic function iswhere

The vertexis located at

### How To

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

1. Identify the horizontal shift of the parabola; this value isIdentify the vertical shift of the parabola; this value is
2. Substitute the values of the horizontal and vertical shift forandin the function
3. Substitute the values of any point, other than the vertex, on the graph of the parabola forand
4. Solve for the stretch factor,
5. Expand and simplify to write in general form.

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

Write an equation for the quadratic functionin (Figure) as a transformation ofand then expand the formula, and simplify terms to write the equation in general form.

We can see the graph of g is the graph ofshifted to the left 2 and down 3, giving a formula in the form

Substituting the coordinates of a point on the curve, such aswe can solve for the stretch factor.

In standard form, the algebraic model for this graph is

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.[/hidden-answer]

#### Analysis

We can check our work using the table feature on a graphing utility. First enterNext, selectthen useandand selectSee (Figure).

 –6 –4 –2 0 2 5 –1 –3 –1 5

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

### Try It

A coordinate grid has been superimposed over the quadratic path of a basketball in (Figure). Find an equation for the path of the ball. Does the shooter make the basket?

The path passes through the origin and has vertex atsoTo make the shot,would need to be about 4 buthe doesn’t make it.

### How To

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

1. Identify
2. Findthe x-coordinate of the vertex, by substitutingandinto
3. Findthe y-coordinate of the vertex, by evaluating

### Finding the Vertex of a Quadratic Function

Find the vertex of the quadratic functionRewrite the quadratic in standard form (vertex form).

Rewriting into standard form, the stretch factor will be the same as thein 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 “” from the general form.

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

#### Analysis

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,and where it occurs,

### Try It

Given the equation write the equation in general form and then in standard form.

in general form;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 formwith a positivevalue isorthe range of a quadratic function written in general form with a negativevalue isor

The range of a quadratic function written in standard formwith a positivevalue isthe range of a quadratic function written in standard form with a negativevalue is

### How To

Given a quadratic function, find the domain and range.

1. Identify the domain of any quadratic function as all real numbers.
2. Determine whetheris positive or negative. Ifis positive, the parabola has a minimum. Ifis negative, the parabola has a maximum.
3. Determine the maximum or minimum value of the parabola,
4. If the parabola has a minimum, the range is given byorIf the parabola has a maximum, the range is given byor

### Finding the Domain and Range of a Quadratic Function

Find the domain and range of

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

Becauseis negative, the parabola opens downward and has a maximum value. We need to determine the maximum value. We can begin by finding thevalue of the vertex.

The maximum value is given by

### Try It

Find the domain and range of

The domain is all real numbers. The range isor

### 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).

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.

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

Let’s use a diagram such as (Figure) to record the given information. It is also helpful to introduce a temporary variable,to represent the width of the garden and the length of the fence section parallel to the backyard fence.

1. We know we have only 80 feet of fence available, andor more simply,This allows us to represent the width,in terms of

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 lengthThe function, written in general form, is

2. 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. Sinceis the coefficient of the squared term,and

To find the vertex:

The maximum value of the function is an area of 800 square feet, which occurs whenfeet. 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.

#### Analysis

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).

### How To

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

1. Write a quadratic equation for a revenue function.
2. Find the vertex of the quadratic equation.
3. 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,for price per subscription andfor quantity, giving us the equation

Because the number of subscribers changes with the price, we need to find a relationship between the variables. We know that currentlyandWe also know that if the price rises to $32, the newspaper would lose 5,000 subscribers, giving a second pair of values,andFrom 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 equationrelating 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.

#### Analysis

This could also be solved by graphing the quadratic as in (Figure). 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 theintercept of a quadratic by evaluating the function at an input of zero, and we find theintercepts at locations where the output is zero. Notice in (Figure) that the number ofintercepts can vary depending upon the location of the graph.

### How To

Given a quadratic functionfind the and x-intercepts.

1. Evaluateto find the y-intercept.
2. Solve the quadratic equationto find the x-intercepts.

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

Find the y– and x-intercepts of the quadratic

We find the y-intercept by evaluating

So the y-intercept is at

For the x-intercepts, we find all solutions of

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

So the x-intercepts are atand

#### Analysis

By graphing the function, we can confirm that the graph crosses the y-axis atWe can also confirm that the graph crosses the x-axis atandSee (Figure)

#### Rewriting Quadratics in Standard Form

In (Figure), 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.

### How To

Given a quadratic function, find theintercepts by rewriting in standard form.

1. Substituteandinto
2. Substituteinto the general form of the quadratic function to find
3. Rewrite the quadratic in standard form usingand
4. Solve for when the output of the function will be zero to find theintercepts.

### Finding the x-Intercepts of a Parabola

Find theintercepts of the quadratic function

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 thatThen we solve forand

So now we can rewrite in standard form.

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

The graph has x-intercepts atand

We can check our work by graphing the given function on a graphing utility and observing theintercepts. See (Figure).

#### Analysis

We could have achieved the same results using the quadratic formula. Identifyand

So the x-intercepts occur atand

### Try It

In a Try It, we found the standard and general form for the functionNow find the y– and x-intercepts (if any).

y-intercept at (0, 13), Nointercepts

### 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 ball’s height above ground can be modeled by the equation

1. When does the ball reach the maximum height?
2. What is the maximum height of the ball?
3. When does the ball hit the ground?

1. The ball reaches the maximum height at the vertex of the parabola.

The ball reaches a maximum height after 2.5 seconds.

2. To find the maximum height, find thecoordinate of the vertex of the parabola.

The ball reaches a maximum height of 140 feet.

3. To find when the ball hits the ground, we need to determine when the height is zero,

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).

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.

### Try It

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 rock’s height above ocean can be modeled by the equation

1. When does the rock reach the maximum height?
2. What is the maximum height of the rock?
3. When does the rock hit the ocean?

3 seconds256 feet7 seconds

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

### Key Equations

 general form of a quadratic function standard form of a quadratic function

### Key Concepts

• A polynomial function of degree two is called a quadratic function.
• The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
• The axis of symmetry is the vertical line passing through the vertex. The zeros, orintercepts, are the points at which the parabola crosses theaxis. Theintercept is the point at which the parabola crosses theaxis. See (Figure), (Figure), and (Figure).
• Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph. See (Figure).
• The vertex can be found from an equation representing a quadratic function. See (Figure).
• The domain of a quadratic function is all real numbers. The range varies with the function. See (Figure).
• A quadratic function’s minimum or maximum value is given by thevalue of the vertex.
• The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. See (Figure) and (Figure).
• The vertex and the intercepts can be identified and interpreted to solve real-world problems. See (Figure).

### Section Exercises

#### Verbal

When written in that form, the vertex can be easily identified.

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

Explain why the condition ofis imposed in the definition of the quadratic function.

Ifthen the function becomes a linear 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?

If possible, we can use factoring. Otherwise, we can use the quadratic formula.

#### Algebraic

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

Vertex

Vertex

Vertex

Vertex

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.

Minimum isand occurs atAxis of symmetry is

Minimum isand occurs atAxis of symmetry is

Minimum isand occurs at Axis of symmetry is

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

Domain isRange is

Domain isRange is

Domain isRange is

For the following exercises, use the vertexand a point on the graphto find the general form of the equation of the quadratic function.

#### Graphical

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

VertexAxis of symmetry isIntercepts are

VertexAxis of symmetry is

VertexAxis of symmetry isIntercepts are

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

#### Numeric

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.

 –2 –1 0 1 2 5 2 1 2 5

 –2 –1 0 1 2 1 0 1 4 9
 –2 –1 0 1 2 –2 1 2 1 –2

 –2 –1 0 1 2 –8 –3 0 1 0
 –2 –1 0 1 2 8 2 0 2 8

#### Technology

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

Graph on the same set of axes the functions

What appears to be the effect of changing the coefficient?

Graph on the same set of axes andand What appears to be the effect of adding a constant?

The graph is shifted up or down (a vertical shift).

Graph on the same set of axes

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 functionwhereis the horizontal distance traveled andis 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.

50 feet

A suspension bridge can be modeled by the quadratic functionwithwhereis the number of feet from the center andis 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.

#### Extensions

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.

Vertexopens up.

Domain isRange is

Vertexopens down.

Vertexopens down.

Domain isRange is

Vertexopens 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.

Containsand has shape ofVertex is on theaxis.

Containsand has the shape ofVertex is on theaxis.

Containsand has the shape ofVertex is on theaxis.

Containsand has the shape ofVertex is on theaxis.

Containsand has the shape ofVertex is on theaxis.

Containshas the shape ofVertex has x-coordinate of

#### Real-World Applications

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

50 feet by 50 feet. Maximize

Find the dimensions of the rectangular corral 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 corral producing the greatest enclosed area split into 3 pens of the same size given 500 feet of fencing.

125 feet by 62.5 feet. Maximize

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?

andproduct is –36; maximize

Suppose that the price per unit in dollars of a cell phone production is modeled bywhereis in thousands of phones produced, and the revenue represented by thousands of dollars isFind 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 byFind the maximum height the rocket attains.

2909.56 meters

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 byHow 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?

\$10.70

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?

### Glossary

axis of symmetry
a vertical line drawn through the vertex of a parabola, that opens up or down, around which the parabola is symmetric; it is defined by
general form of a quadratic function
the function that describes a parabola, written in the form, whereandare real numbers and
roots
in a given function, the values ofat which, also called zeros
standard form of a quadratic function
the function that describes a parabola, written in the form, whereis the vertex
vertex
the point at which a parabola changes direction, corresponding to the minimum or maximum value of the quadratic function
vertex form of a quadratic function
another name for the standard form of a quadratic function
zeros
in a given function, the values ofat which, also called roots