Question

x

y

----

----

$-\frac{pi}{2}$

0

0

6

$\frac{pi}{2}$

0

$pi$

-6

$\frac{3pi}{2}$

0

the function represented by the table has the form $y = acos(bx)$. which of the following statements are true?choose three correct answers.1. the maximum value of the function is 6.2. the frequency is $\frac{1}{2pi}$3. the period is $pi$.4. the function increases on the interval $(pi, 2pi)$.5. the graph is compressed horizontally by a factor of $\frac{1}{2}$

Explanation:

Step1: Identify amplitude $a$

From the table, the maximum $y$-value is 6, so $a=6$.

Step2: Find period $T$

The function goes from maximum (at $x=0$) to minimum (at $x=\pi$) and back, so the full period is $2\pi$.

Step3: Calculate $b$ from period

Use $T=\frac{2\pi}{b}$, so $b=\frac{2\pi}{T}=\frac{2\pi}{2\pi}=1$. The function is $y=6\cos(x)$.

Step4: Verify each statement

1. Max value is 6: True (matches $a=6$).
2. Frequency is $\frac{1}{2\pi}$: Frequency $=\frac{1}{T}=\frac{1}{2\pi}$, True.
3. Period is $\pi$: False (period is $2\pi$).
4. Increases on $(\pi,2\pi)$: $\cos(x)$ increases on $(\pi,2\pi)$, so $6\cos(x)$ does too, True.
5. Horizontal compression by $\frac{1}{2}$: $b=1$, no compression, False.

Answer:

• The maximum value of the function is 6.
• The frequency is $\frac{1}{2\pi}$.
• The function increases on the interval $(\pi, 2\pi)$.