QuestionThe points $(-3,-4)$ and $(-5,-9)$ are a maximum and minimum, respectively, of a periodic function $f(x)$ , which has period 9 . What is the amplitude of the function? The amplitude is $\square$ 2.5 $\square$ What is an equation for the midline? The midline is $y=$ $\square$ $-6.5$
Which of the following points must lie on the graph of the function $y=f(x)$ ? Select all that are correct. $(31,-12)$ $(-50,-9)$ $(22,-4)$ $(31,-9)$ $(24,-4)$ $(-48,-1)$ None of the above

Studdy Solution

STEP 1

1. The function $f(x)$ is periodic with a period of 9.
2. The points $(-3, -4)$ and $(-5, -9)$ are the maximum and minimum points, respectively.
3. We need to find the amplitude and the equation for the midline.
4. We need to determine which points must lie on the graph of $y = f(x)$ .

STEP 2

1. Calculate the amplitude of the function.
2. Determine the equation for the midline.
3. Identify points that must lie on the graph of $y = f(x)$ .

STEP 3

The amplitude of a periodic function is half the difference between the maximum and minimum values.
Calculate the amplitude: $\text{Amplitude} = \frac{\text{Maximum} - \text{Minimum}}{2} = \frac{-4 - (-9)}{2} = \frac{5}{2} = 2.5$

STEP 4

The midline of a periodic function is the average of the maximum and minimum values.
Calculate the midline: $\text{Midline} = \frac{\text{Maximum} + \text{Minimum}}{2} = \frac{-4 + (-9)}{2} = \frac{-13}{2} = -6.5$

STEP 5

Since the function is periodic with a period of 9, any point $(x, y)$ on the graph will repeat every 9 units.
Check each point: - $(31, -12)$ : Not on the graph since $-12$ is not between $-4$ and $-9$ . - $(-50, -9)$ : Check if $(-5, -9)$ repeats every 9 units: $-5 - 9k = -50$ for some integer $k$ . Solving gives $k = 5$ , so $(-50, -9)$ is on the graph. - $(22, -4)$ : Check if $(-3, -4)$ repeats every 9 units: $-3 + 9k = 22$ for some integer $k$ . Solving gives $k = 3$ , so $(22, -4)$ is on the graph. - $(31, -9)$ : Check if $(-5, -9)$ repeats every 9 units: $-5 + 9k = 31$ for some integer $k$ . Solving gives $k = 4$ , so $(31, -9)$ is on the graph. - $(24, -4)$ : Check if $(-3, -4)$ repeats every 9 units: $-3 + 9k = 24$ for some integer $k$ . Solving gives $k = 3$ , so $(24, -4)$ is on the graph. - $(-48, -1)$ : Not on the graph since $-1$ is not between $-4$ and $-9$ .
The amplitude is $\boxed{2.5}$ .
The midline is $y = \boxed{-6.5}$ .
The points that must lie on the graph are: $(-50, -9)$ , $(22, -4)$ , $(31, -9)$ , $(24, -4)$ .

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