Question

Compare the functions shown below:

Answer 1

We have been given three functions. We are asked to find the function which has the the greatest y-intercept.

We know that y-intercept of a function is the point, when the graph crosses y-axis. We know that the value of x is 0 on y-axis.

Upon looking at given graph of function $g(x)$, we can see that it crosses y-axis at $(0,-3)$, therefore, the y-intercept of the function $g(x)$ is $-3$.

We can see that function $f(x)$ is written in slope intercept form.

We know that slope-intercept form of an equation is $y=mx+b$, where, m represents slope of line and b represents the y-intercept.

$f(x)=-3x+2$

Upon looking at  $f(x)$, we can see that the value of b is 2, therefore, the y-intercept of  $f(x)$ is 2.

To find y-intercept of $h(x)$, we will substitute $x=0$ in function and solve as:

$h(x)=4\text{sin}(2x+\pi)+3$

$h(0)=4\text{sin}(2(0)+\pi)+3$

$h(0)=4\text{sin}(0+\pi)+3$

$h(0)=4\text{sin}(\pi)+3$

$h(0)=4(0)+3$

$h(0)=3$

Therefore, the y-intercept of $h(x)$ is 3.

Since 3 is greater than 2 and $-3$, therefore, $h(x)$ has the greatest y-intercept.

Answer 2

Find all y-intercepts:

1. For f(x): when x=0 you have that f(0)=-3·0-5=-5, then y-intercept is a point (0,-5).

2. For g(x): from the task conditions you have that g(x) is cosine function with y intercept at 0, negative 3, then y-intercept is a point (0,-3).

3. For h(x): when x=0 you have that h(0)=cos(2·0-π)+4=cos(-π)+4=-1+4=3, then y-intercept is a point (0,3).

As you can see the greatest y-intercept has function h(x).

Answer: correct choice is C.