Question

1. is the following function linear or nonlinear? how do you know?
2. what is the equation for the above function? show all your work.

Explanation:

Step1: Check linearity via graph

A linear function forms a straight line. The given points form a curve, not a straight line, so it is nonlinear.

Step2: Identify function type

The points follow a pattern where the change in y slows as x increases, matching a square root function form $y = a\sqrt{x + h} + k$. Use the point $(0,5)$:
$5 = a\sqrt{0 + h} + k$
Use the point $(-9,0)$:
$0 = a\sqrt{-9 + h} + k$
Assume $h=9$ to eliminate the square root in the second equation:
$0 = a\sqrt{-9 + 9} + k \implies k=0$
Substitute $k=0$ and $(0,5)$:
$5 = a\sqrt{0 + 9} \implies 5 = 3a \implies a=\frac{5}{3}$
Verify with another point, e.g., $x=-5$:
$y=\frac{5}{3}\sqrt{-5 + 9}=\frac{5}{3}\times2=\frac{10}{3}\approx3.33$, which matches the plotted point.

Answer:

1. The function is nonlinear. A linear function graphs as a straight line, but these points form a curved path, so it does not have a constant rate of change.
2. The equation of the function is $\boldsymbol{y=\frac{5}{3}\sqrt{x+9}}$