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{downarrow}) 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 ( \boldsymbol{square} ). - for ( x ) between ( x = pi ) and ( x = 2pi ), the function value ( y \boldsymbol{downarrow} ) - for ( x ) between ( x = pi ) and ( x = 2pi ), the function value ( y \boldsymbol{downarrow} 0 ).

Explanation:

Step1: Identify function type

The graph is periodic, symmetric about the y-axis, matches the shape of $y=\cos(x)$, so it is a cosine (even, periodic) function.

Step2: Find y-intercept

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

Step3: Find smallest positive x-intercept

The x-intercepts are where $y=0$. The smallest positive one is at $x=\frac{\pi}{2}$.

Step4: Find maximum y and its x

The highest point on the graph is $y=1$, which occurs at $x=0, \pm2\pi, \pm4\pi,...$; the simplest value is $x=0$.

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

Looking at the graph, between $x=\pi$ and $x=2\pi$, the function values are above the x-axis, so $y \geq 0$ (specifically, $y > 0$ except at the endpoints).

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 $x = \frac{\pi}{2}$
• 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 $y \geq 0$ (or $y > 0$ for non-endpoint values)