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
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: $(-50, -9)$, $(22, -4)$, $(31, -9)$, $(24, -4)$.