Question

3
compare the rates of change of the following table and graph.

x	y
2	-8
3	-12
4	-16

(graph of a line on a coordinate plane with x-axis from -4 to 4 and y-axis from -4 to 4, passing through points, with a red line)

a. the rate of change of the graph is greater than the rate of change of the table.

b. the rate of change of the table is greater than the rate of change of the graph.

c. the rate of change of the table is equal to the rate of change of the graph.

d. there is not enough information to determine the rates of change.

Explanation:

Step1: Calculate rate of change for the table

The rate of change (slope) for a table with two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(\frac{y_2 - y_1}{x_2 - x_1}\). Taking two points from the table, say \((1, -4)\) and \((2, -8)\):
\(\frac{-8 - (-4)}{2 - 1}=\frac{-4}{1}=-4\)

Step2: Calculate rate of change for the graph

The graph is a line. Let's find two points on the graph. From the graph, it passes through \((0, 0)\) and \((1, -4)\) (by observing the grid). The rate of change (slope) is \(\frac{-4 - 0}{1 - 0}=-4\)

Step3: Compare the two rates of change

The rate of change of the table is \(-4\) and the rate of change of the graph is also \(-4\). So they are equal.

Answer:

C. The rate of change of the table is equal to the rate of change of the graph.