Question

which of the relationships below represents a function with a greater slope than the function $y = -2x - 2?$

Explanation:

Step1: Identify reference slope

The given function $y=-2x-2$ is in slope-intercept form $y=mx+b$, so its slope $m_0=-2$. We need a slope $m$ where $m < -2$ (steeper negative slope) or $m > -2$? Wait, no: "greater slope" means comparing numerical values. A slope is greater if it is more positive. Wait, no: let's calculate each option's slope:

Step2: Calculate slope for Option A

Pick two points: $(0,5)$ and $(1,0)$.
Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m_A=\frac{0-5}{1-0}=-5$

Step3: Calculate slope for Option B

Pick two points: $(0,5)$ and $(1,-1)$.
$m_B=\frac{-1-5}{1-0}=-6$

Step4: Calculate slope for Option C

Use points $(-1,-3)$ and $(2,-9)$.
$m_C=\frac{-9-(-3)}{2-(-1)}=\frac{-6}{3}=-2$

Step5: Calculate slope for Option D

Use points $(0,-5)$ and $(4,3)$.
$m_D=\frac{3-(-5)}{4-0}=\frac{8}{4}=2$

Step6: Compare slopes to $-2$

We need a slope greater than $-2$. Compare numerical values:
$-5 < -2$, $-6 < -2$, $-2 = -2$, $2 > -2$

Answer:

D. The function with the table:
$x$: $-4, 0, 4, 8$; $y$: $-13, -5, 3, 11$