To determine the constant rate of change between the points, we need to check how the output values (y-values) change relative to the input values (x-values). In this case, the given points are:

• (1, 2)
• (2, 8)
• (3, 32)
• (4, 128)

Step 1: Check the ratio or slope between consecutive points.

The rate of change between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$\text{Rate of Change} = \frac{y_2 - y_1}{x_2 - x_1}$

Now, let’s compute the rate of change between consecutive points.

Between (1, 2) and (2, 8):

$\frac{8 - 2}{2 - 1} = \frac{6}{1} = 6$

Between (2, 8) and (3, 32):

$\frac{32 - 8}{3 - 2} = \frac{24}{1} = 24$

Between (3, 32) and (4, 128):

$\frac{128 - 32}{4 - 3} = \frac{96}{1} = 96$

Step 2: Is the rate of change constant?

Since the rate of change between consecutive points is not the same (6, 24, 96), this graph does not have a constant rate of change.

Observation:

The outputs are growing exponentially rather than linearly. Specifically:

• $2, 8, 32, 128$ follow a pattern: $y = 2^{x+1}$.

This suggests the function is exponential, not linear.

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Do you have any follow-up questions or need further clarification?

Here are 5 questions to extend the topic:

1. How do you determine the slope of a linear function?
2. What is the difference between a linear and exponential function?
3. How can we find the equation for an exponential function from points?
4. What does it mean for a function to have a "constant rate of change"?
5. How do exponential functions appear on a graph compared to linear functions?

Tip: If the ratio of change between consecutive $y$-values grows by a consistent factor (instead of a consistent difference), the function is likely exponential.