Q12.

Question

Consider the given function.

$y = - x^{2} + 3 x$

a. Determine whether the function has a maximum or minimum value.

b. State the maximum or minimum value.

c. What are the domain and range of the function?

Step-by-Step Solution

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Answer

a. The function $y = - x^{2} + 3 x$ has a maximum value.

b. The maximum value of the function $y = - x^{2} + 3 x$ is $\frac{9}{4}$ .

c. The domain is $(- \infty, \infty)$ and the range is $(- \infty, \frac{9}{4}]$ .

1Part a. Step 1. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $y = a x^{2} + b x + c$ , where, $a \neq 0$ is called the standard form of the quadratic function.

2Part a. Step 2. Define the maximum or minimum point of the function .

The graph of the function $y = a x^{2} + b x + c,$

Opens upward and has a minimum value at $x = - \frac{b}{2 a}$ , when $a > 0$ .

Opens downward and has a maximum value at $x = - \frac{b}{2 a}$ , when $a < 0$ .

3Part a. Step 3. Determine whether the function y = − x 2 + 3 x has a maximum or minimum value.

Compare the quadratic function $y = - x^{2} + 3 x$ with the standard quadratic function $y = a x^{2} + b x + c$ .

$a = - 1, b = 3, c = 0$

Since $a < 0$ .

Hence, the graph of the function $y = - x^{2} + 3 x$ opens downward and has a maximum value.

Therefore, the function width="91" style="max-width: none; vertical-align: -4px;" $y = - x^{2} + 3 x$ has a maximum value.

4Part b. Step 1. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $y = a x^{2} + b x + c$ , where, $a \neq 0$ is called the standard form of the quadratic function.

5Part b. Step 2. Define the maximum or minimum point of the function y = a x 2 + b x + c .

The graph of the function $y = a x^{2} + b x + c,$

Opens upward and has a minimum value at $x = - \frac{b}{2 a}$ , when $a > 0$ .

Opens downward and has a maximum value at $x = - \frac{b}{2 a}$ , when $a < 0$ .

6Part b. Step 3. Calculate the maximum or minimum value of the function y = − x 2 + 3 x .

Compare the quadratic function $y = - x^{2} + 3 x$ with the standard quadratic function $y = a x^{2} + b x + c$ .

$a = - 1, b = 3, c = 0$

Substitute $a = - 1$ and $b = 3$ in $x = - \frac{b}{2 a}$ .

$x = - (\frac{3}{2 (- 1)}) x = - (- \frac{3}{2}) x = \frac{3}{2}$

Since $a < 0$ .

Hence, the graph of the function $y = - x^{2} + 3 x$ opens downward and has a maximum value at $x = \frac{3}{2}$ .

Substitute $x = \frac{3}{2}$ in $y = - x^{2} + 3 x$ .

$y = - {(\frac{3}{2})}^{2} + 3 (\frac{3}{2}) y = - \frac{9}{4} + \frac{9}{2} y = \frac{- 9 + 18}{4} y = \frac{9}{4}$

Therefore the maximum value of the function $y = - x^{2} + 3 x$ is $\frac{9}{4}$ .

7Part c. Step 1. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $y = a x^{2} + b x + c$ , where, $a \neq 0$ is called the standard form of the quadratic function.