Question

which of these graphs represents a rate of change of 4?

Explanation:

Step1: Recall the formula for slope (rate of change)

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For a line, the rate of change is the slope of the line.

Step2: Analyze the first graph

Let's assume two points on the first - graph. If we take two points \((x_1,y_1)=(-3,0)\) and \((x_2,y_2)=(-2,-4)\). Then \(m=\frac{-4 - 0}{-2+3}=\frac{-4}{1}=- 4\).

Step3: Analyze the second graph

Let's take two points \((x_1,y_1)=(0,-3)\) and \((x_2,y_2)=(4,-2)\). Then \(m=\frac{-2 + 3}{4-0}=\frac{1}{4}\).

Step4: Analyze the third graph

Let's assume two points \((x_1,y_1)=(-4,0)\) and \((x_2,y_2)=(4,-2)\). Then \(m=\frac{-2-0}{4 + 4}=\frac{-2}{8}=-\frac{1}{4}\).

Answer:

The first graph (the one with a steeper negative - looking line) has a rate of change (slope) of \(-4\), the second graph has a rate of change (slope) of \(\frac{1}{4}\), and the third graph has a rate of change (slope) of \(-\frac{1}{4}\). Since the problem is likely missing some options (assuming the first graph is the one with a steeper line which has a magnitude of slope \(4\) in the negative direction. If we consider the absolute value, and if there was a graph with a positive slope of \(4\) (not shown in the provided images but based on the formula application), the slope (rate of change) formula \(m = \frac{y_2-y_1}{x_2 - x_1}\) is used to calculate the rate of change for a linear function \(y=mx + b\) where \(m\) is the rate of change.