Question

the illustration below shows the graph of $y$ as a function of $x$.
complete the following sentences based on the graph of the function.

• this is the graph of a $\boldsymbol{

abla}$ function.

• the $y$-intercept of the graph is the function value $y = \boldsymbol{square}$.
• the smallest positive $x$-intercept of the graph is located at $x = \boldsymbol{square}$.
• the greatest value of $y$ is $y = \boldsymbol{square}$, and it occurs when $x = \boldsymbol{square}$.
• for $x$ between $x = \pi$ and $x = 2\pi$, the function value $y$ $\boldsymbol{

abla}$ 0.

Explanation:

Step1: Identify function type

The graph is periodic, symmetric about the y-axis, has a maximum at x=0, and matches the shape of $y=\cos(x)$, so it is a cosine (or cosine-like periodic/even) function.

Step2: Find y-intercept

The y-intercept is where $x=0$. From the graph, $y=1$ at $x=0$.

Step3: Find smallest positive x-intercept

The x-intercepts are where $y=0$. The smallest positive x-value here is $x=\pi$.

Step4: Find maximum y and its x

The highest point on the graph is $y=1$, which occurs at $x=0$.

Step5: Analyze y between $\pi$ and $2\pi$

Between $x=\pi$ and $x=2\pi$, the graph is below the x-axis, so $y < 0$.

Answer:

• This is the graph of a cosine (even/periodic) function.
• The y-intercept of the graph is the function value $y=1$.
• The smallest positive x-intercept of the graph is located at $x=\pi$.
• The greatest value of $y$ is $y=1$, and it occurs when $x=0$.
• For $x$ between $x=\pi$ and $x=2\pi$, the function value $y < 0$.