Problem Breakdown

The question asks us to determine which of the given relationships has a greater rate of change than the function shown on the graph. Here's how we'll solve the problem:

1. Identify the slope of the graphed function:
   The function is a straight line, so its rate of change (slope) can be calculated using the formula for slope:
   $m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

   From the graph:

   • Two points on the line are $(-6, 6)$ and $(6, -6)$.
   • Using the slope formula: $m = \frac{-6 - 6}{6 - (-6)} = \frac{-12}{12} = -1$ So, the slope of the graphed function is -1.

2. Analyze each option's rate of change:
   We will now compute the slope for each option.

Option A (Table)

Using $(x_1, y_1) = (4, -12)$ and $(x_2, y_2) = (8, -19)$: $m = \frac{-19 - (-12)}{8 - 4} = \frac{-19 + 12}{4} = \frac{-7}{4} = -1.75$ Slope = -1.75 (greater than -1 in magnitude)

Option B (Table)

Using $(x_1, y_1) = (5, -11)$ and $(x_2, y_2) = (9, -23)$: $m = \frac{-23 - (-11)}{9 - 5} = \frac{-23 + 11}{4} = \frac{-12}{4} = -3$ Slope = -3 (greater than -1 in magnitude)

Option C ($y = -\frac{5}{4}x - 2$)

The slope is the coefficient of $x$:
$m = -\frac{5}{4} = -1.25$ Slope = -1.25 (greater than -1 in magnitude)

Option D ($y = \frac{1}{2}x - 2$)

The slope is the coefficient of $x$:
$m = \frac{1}{2} = 0.5$ Slope = 0.5 (less than -1 in magnitude)

Correct Answer:

The relationships with a greater rate of change (in absolute value) than the graphed function are A, B, and C. However, the question likely wants the greatest, which is B (with a slope of -3).