Lesson 23

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?

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?
\(f(x)=\text-(x-4)^2+6\)	\((4,6)\)
\(g(x)=(x+7)^2-3\)	\((\text-7,\text-3)\)
\(h(x)=4(x+5)^2+7\)	\((\text-5,7)\)
\(k(x)=x^2-6x-3\)	\((3,\text-12)\)
\(m(x)=\text-x^2+8x\)	\((4,16)\)

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.

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 \(g(x)=x^2+6x-1\).

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 \(x=\text{-}3\).

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 \(f\).