Question

function h is the sum of 3 and four times the cube of x.
this function has the highest y - intercept.
this function is increasing over the longest interval.
this function has the lowest y - intercept.
(the table has x values - 2, - 1, 0, 1, 2 and corresponding g(x) values 8, 0.5, - 1, - 2.5, - 10; there is also a graph of a function on a coordinate grid)

Explanation:

Step1: Find y-intercept of first function

The y-intercept is where $x=0$. From the graph, the curve crosses the y-axis at $(0,0)$, so y-intercept = $0$.

Step2: Find y-intercept of $g(x)$

From the table, when $x=0$, $g(0)=-1$, so y-intercept = $-1$.

Step3: Find y-intercept of $h(x)$

First, write $h(x)$: $h(x)=4x^3+3$. At $x=0$, $h(0)=4(0)^3+3=3$, so y-intercept = $3$.

Step4: Compare y-intercepts

Order: $3 > 0 > -1$. So $h(x)$ has highest, $g(x)$ has lowest.

Step5: Analyze increasing intervals

First function (graph): It decreases on $(-\infty,0)$ and increases on $(0,\infty)$, so increasing interval is $(0,\infty)$.
$g(x)$: As $x$ increases from $-2$ to $2$, $g(x)$ decreases from $8$ to $-10$, so it is always decreasing.
$h(x)$: The derivative $h'(x)=12x^2\geq0$ for all real $x$, so it is increasing on $(-\infty,\infty)$, which is the longest interval.

Answer:

• "This function has the highest y-intercept." → Function $h(x)$ (sum of 3 and four times the cube of $x$)
• "This function is increasing over the longest interval." → Function $h(x)$ (sum of 3 and four times the cube of $x$)
• "This function has the lowest y-intercept." → Function $g(x)$ (the table function)