Math Snap

STEP 1

What is this asking?
We need to find how much the graph changes vertically (net change) and at what rate it changes on average between two points.
Watch out!
Don't mix up net change and average rate of change; net change is just the difference in the y-values, while the average rate of change also considers the difference in x-values.

STEP 2

1. Find the Net Change
2. Find the Average Rate of Change

STEP 3

Let's define our two points!
We'll call the first point $(x_1, y_1) \approx (1, 2)$ and the second point $(x_2, y_2) \approx (4, 6)$.
Remember, the x-values represent the horizontal position, and the y-values represent the vertical position.

STEP 4

To find the net change, we simply subtract the initial y-value from the final y-value.
This tells us how much the graph has moved vertically between the two points.

STEP 5

$$y_2 - y_1 = 6 - 2 = 4$$ So, our net change is 4!
The graph has moved up 4 units between the two points.

STEP 6

The average rate of change tells us how much the graph changes vertically for every one unit it changes horizontally.
It's like finding the slope of a line between the two points.

STEP 7

To calculate this, we take the net change (which we already found!) and divide it by the change in the x-values.
This gives us the average steepness of the graph between the two points.

STEP 8

$$\frac{y_2 - y_1}{x_2 - x_1} = \frac{6 - 2}{4 - 1} = \frac{4}{3}$$ So, our average rate of change is $\frac{4}{3}$!
This means that, on average, for every 1 unit we move to the right, the graph moves up $\frac{4}{3}$ units.

(a) The net change between the points is 4.
(b) The average rate of change between the points is $\frac{4}{3}$.