Q15.

Question

A baseball is thrown with an upward velocity of 32 feet per second. The equation $h = - 16 t^{2} + 32 t$ gives the height of the ball t seconds after it is thrown.

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

b. State the maximum or minimum value.

c. State a reasonable domain and range for this situation?

Step-by-Step Solution

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Answer

a. The function $h = - 16 t^{2} + 32 t$ has a maximum value.

b. The maximum value of the function is $h = - 16 t^{2} + 32 t$ 16.

c. The domain is $(- \infty, \infty)$ and the range is $(- \infty, 16]$ .

1Step 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.

2Step 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 $a < 0$ .

3Step3. calculation

a. Compare the quadratic function $h = - 16 t^{2} + 32 t$ with the standard quadratic function $h = a t^{2} + b t + c$ .

$a = - 16, b = 32, c = 0$

Since $a < 0$ .

Hence, the graph of the function $h = - 16 t^{2} + 32 t$ opens downward and has a maximum value.

Therefore, the function $h = - 16 t^{2} + 32 t$ has a maximum value.

b. Compare the quadratic function $h = - 16 t^{2} + 32 t$ with the standard quadratic function $h = a t^{2} + b t + c$ .

$a = - 16, b = 32, c = 0$

Substitute $a = - 16$ and $b = 32$ in $t = - \frac{b}{2 a}$ .

$t = - (\frac{32}{2 (- 16)}) t = - (- 1) t = 1$

Since $a < 0$ .

Hence, the graph of the function $h = - 16 t^{2} + 32 t$ opens downward and has a maximum value at $t = 1$ .

Substitute $t = 1$ in $h = - 16 t^{2} + 32 t$ .

$h = - 16 (1^{2}) + 32 (1) h = - 16 + 32 h = 16$

Therefore the maximum value of the function $h = - 16 t^{2} + 32 t$ is 16.

c. The domain is the set of all of the possible values of the independent variable $x$ .

The range is the set of all the possible values of the dependent variable $y$ .

Since the graph of the function $h = - 16 t^{2} + 32 t$ is a parabola.

Since the parabola always extends to infinity.

So, the domain is $(- \infty, \infty)$ .

Since the maximum value of the function is 2.

So, the range is $(- \infty, 16]$ .

Therefore, the domain is $(- \infty, \infty)$ and the range is $(- \infty, 16]$ .

Q14.

Q16.

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