Lesson 23

Using Quadratic Expressions in Vertex Form to Solve Problems

  • Let’s find the maximum or minimum value of a quadratic function.

23.1: Values of a Function

Here are graphs that represent two functions, \(f\) and \(g\), defined by:

\(f(x)=(x-4)^2+1\)

\(g(x)=\text-(x-12)^2+7\)

Function f and g on a grid.
  1. \(f(1)\) can be expressed in words as “the value of \(f\) when \(x\) is 1.” Find or compute:

    1. the value of \(f\) when \(x\) is 1
    2. \(f(3)\)
    3. \(f(10)\)
  2. Can you find an \(x\) value that would make \(f(x)\):

    1. Less than 1?
    2. Greater than 10,000?
  3. \(g(9)\) can be expressed in words as “the value of \(g\) when \(x\) is 9.” Find or compute:

    1. the value of \(g\) when \(x\) is 9
    2. \(g(13)\)
    3. \(g(2)\)
  4. Can you find an \(x\) value that would make \(g(x)\):

    1. Greater than 7?
    2. Less than -10,000?

23.2: Maximums and Minimums

  1. The graph that represents \(p(x)=(x-8)^2+1\) has its vertex at \((8,1)\). Here is one way to show, without graphing, that \((8,1)\) corresponds to the minimum value of \(p\).

    • When \(x=8\), the value of \((x-8)^2\) is 0, because \((8-8)^2 = 0^2 = 0\).
    • Squaring any number always results in a positive number, so when \(x\) is any value other than 8, \((x-8)\) will be a number other than 0, and when squared, \((x-8)^2\) will be positive.
    • Any positive number is greater than 0, so when \(x \neq 8\), the value of \((x-8)^2\) will be greater than when \(x= 8\). In other words, \(p\) has the least value when \(x=8\).

    Use similar reasoning to explain why the point \((4,1)\) corresponds to the maximum value of \(q\), defined by \(q(x) = \text-2(x-4)^2+1\).

  2. Here are some quadratic functions, and the coordinates of the vertex of the graph of each. Determine if the vertex corresponds to the maximum or the minimum value of the function. Be prepared to explain how you know.

equation coordinates of
the vertex
maximum or minimum?
\(f(x)=\text-(x-4)^2+6\) \((4,6)\)  
\(g(x)=(x+7)^2-3\) \((\text-7,\text-3)\)  
\(h(x)=4(x+5)^2+7\) \((\text-5,7)\)  
\(k(x)=x^2-6x-3\) \((3,\text-12)\)  
\(m(x)=\text-x^2+8x\) \((4,16)\)  


Here is a portion of the graph of function \(q\), defined by \(q(x)=\text-x^2+14x-40\).

Graph of a function surrounded by rectangle A B C D. Points A and B coincide with the x-intercepts of the graph, and segment C D just touches the vertex of the graph.

\(ABCD\) is a rectangle. Points \(A\) and \(B\) coincide with the \(x\)-intercepts of the graph, and segment \(CD\) just touches the vertex of the graph.

Find the area of \(ABCD\). Show your reasoning.

23.3: All the World’s a Stage

A function \(A\), defined by \(p(600-75p)\), describes the revenue collected from the sales of tickets for Performance A, a musical.

The graph represents a function \(B\) that models the revenue collected from the sales of tickets for Performance B, a Shakespearean comedy.

Graph of function B on a grid.

 

In both functions, \(p\) represents the price of one ticket, and both revenues and prices are measured in dollars.

Without creating a graph of \(A\), determine which performance gives the greater maximum revenue when tickets are \(p\) dollars each. Explain or show your reasoning.

Summary

Any quadratic function has either a maximum or a minimum value. We can tell whether a quadratic function has a maximum or a minimum by observing the vertex of its graph.

Here are graphs representing functions \(f\) and \(g\), defined by \(f(x)=\text-(x+5)^2+4\) and \(g(x)=x^2+6x-1\).

Graph of function f on a grid. X axis from negative 8 to 1, by 1’s. Y axis from negative 6 to 4, by 2’s. Origin, O. Parabola opens downward with vertex at negative 35 comma 4.
  • The vertex of the graph of \(f\) is \((\text-5,4)\) and the graph is a U shape that opens downward.
  • No other points on the graph of \(f\) (no matter how much we zoom out) are higher than \((\text-5,4)\), so we can say that \(f\) has a maximum of 4, and that this occurs when \(x=\text-5\).
Graph of function g on a grid. X axis from negative 8 to 1, by 1’s. Y axis from negative 16 to 4, by 4’s. Origin, O. Graph opens upward with vertex at negative 3 comma negative 10.
  • The vertex of the graph of \(g\) is at \((\text-3, \text-10)\) and the graph is a U shape that opens upward.
  • No other points on the graph (no matter how much we zoom out) are lower than \((\text-3,\text-10)\), so we can say that \(g\) has a minimum of -10, and that this occurs when \(x = \text-3\).

We know that a quadratic expression in vertex form can reveal the vertex of the graph, so we don’t actually have to graph the expression. But how do we know, without graphing, if the vertex corresponds to a maximum or a minimum value of a function?

The vertex form can give us that information as well!

To see if \((\text-3, \text-10)\) is a minimum or maximum of \(g\), we can rewrite \(x^2+6x-1\) in vertex form, which is \((x+3)^2-10\). Let’s look at the squared term in \((x+3)^2-10\).

  • When \(x=\text-3\), \((x+3)\) is 0, so \((x+3)^2\) is also 0.
  • When \(x\) is not -3, the expression \((x+3)\) will be a non-zero number, and \((x+3)^2\) will be positive (squaring any number gives a positive result).
  • Because a squared number cannot have a value less than 0, \((x+3)^2\) has the least value when \(x=3\).

To see if \((\text-5,4)\) is a minimum or maximum of \(f\), let’s look at the squared term in \(\text-(x+5)^2+4\).

  • When \(x=\text-5\), \((x+5)\) is 0, so \((x+5)^2\) is also 0.
  • When \(x\) is not -5, the expression \((x+5)\) will be non-zero, so \((x+5)^2\) will be positive. The expression \(\text-(x+5)^2\) has a negative coefficient of -1, however. Multiplying \((x+5)^2\) (which is positive when \(x \neq \text-5\)) by a negative number results in a negative number.
  • Because a negative number is always less than 0, the value of \(\text-(x+5)^2+4\) will always be less when \(x \neq \text-5\) than when \(x=\text-5\). This means \(x =\text-5\) gives the greatest value of \(f\).

Glossary Entries

  • maximum

    A maximum of a function is a value of the function that is greater than or equal to all the other values. The maximum of the graph of the function is the corresponding highest point on the graph.

  • minimum

    A minimum of a function is a value of the function that is less than or equal to all the other values. The minimum of the graph of the function is the corresponding lowest point on the graph.

  • vertex form (of a quadratic expression)

    The vertex form of a quadratic expression in \(x\) is \(a(x-h)^2 + k\), where \(a\), \(h\), and \(k\) are constants, and \(a\) is not 0.