Applications of Quadratic Functions

Learning Outcomes

  • Solve problems involving the roots and intercepts of a quadratic function.
  • Solve problems involving a quadratic function’s minimum or maximum value.

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

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.

Two graphs where the first graph shows the maximum value for f(x)=(x-2)^2+1 which occurs at (2, 1) and the second graph shows the minimum value for g(x)=-(x+3)^2+4 which occurs at (-3, 4).

Figure 9

Example

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.

Diagram of the garden and the backyard.

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

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

  1. Write a quadratic equation for revenue.
  2. Find the vertex of the quadratic equation.
  3. Determine the y-value of the vertex.

Example

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?

 

Example

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 [latex]H\left(t\right)=-16{t}^{2}+80t+40[/latex].

a. When does the ball reach the maximum height?

b. What is the maximum height of the ball?

c. When does the ball hit the ground?

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 [latex]H\left(t\right)=-16{t}^{2}+96t+112[/latex].

a. When does the rock reach the maximum height?

b. What is the maximum height of the rock?

c. When does the rock hit the ocean?

Applying

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.

Two graphs where the first graph shows the maximum value for f(x)=(x-2)^2+1 which occurs at (2, 1) and the second graph shows the minimum value for g(x)=-(x+3)^2+4 which occurs at (-3, 4).

Example: 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 [latex]L[/latex].
  2. What dimensions should she make her garden to maximize the enclosed area?

The problem we solved above is called a constrained optimization problem. We can optimize our desired outcome given a constraint, which in this case was a limited amount of fencing materials. Try it yourself in the next problem.

Try It

 

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

  1. Write a quadratic equation for revenue.
  2. Find the vertex of the quadratic equation.
  3. Determine the [latex]y[/latex]-value of the vertex.

Example: 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?

In the example above, we knew the number of subscribers to a newspaper and used that information to find the optimal price for each subscription. What if the price of subscriptions is affected by competition?

try it

Previously, we found a quadratic function that modeled revenue as a function of price.

[latex]\text{Revenue}-2,500{p}^{2}+159,000p[/latex]

We found that selling the paper at [latex]\$31.80[/latex] per subscription would maximize revenue.  What if your closest competitor sells their paper for [latex]\$25.00[/latex] per subscription? What is the maximum revenue you can make you sell your paper for the same?

Try It

Key Equations

The quadratic formula [latex]x=\frac{-b\pm \sqrt{{b}^{2}-4ac}}{2a}[/latex]

The discriminant is defined as [latex]b^2-4ac[/latex]

Key Concepts

  • The zeros, or [latex]x[/latex]-intercepts, are the points at which the parabola crosses the [latex]x[/latex]-axis. The [latex]y[/latex]-intercept is the point at which the parabola crosses the [latex]y[/latex]axis.
  • The vertex can be found from an equation representing a quadratic function.
  • A quadratic function’s minimum or maximum value is given by the [latex]y[/latex]-value of the vertex.
  • The minium 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.
  • Some quadratic equations must be solved by using the quadratic formula.
  • The vertex and the intercepts can be identified and interpreted to solve real-world problems.
  • Some quadratic functions have complex roots.

Glossary

discriminant
the value under the radical in the quadratic formula, [latex]b^2-4ac[/latex], which tells whether the quadratic has real or complex roots
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 of [latex]x[/latex] at which [latex]y=0[/latex], also called roots