Analyzing Rate of Change: Function Table from (-2,3) to (4,12)

Q: What's the difference between uniform and non-uniform rate of change?

Uniform rate means the slope is the same between every pair of consecutive points - this creates a straight line. Non-uniform rate means the slopes are different, creating a curved line.

Q: Why can't I just find the slope from the first point to the last point?

That only gives you the average rate of change over the entire interval! To check if it's uniform, you need to see if the rate is constant throughout by checking each step separately.

Q: In this problem, why are the slopes 1, 3/2, and 2 all different?

The y-values increase by 2, then 3, then 4 while x increases by 2 each time. Since the y-changes aren't constant, the slopes $ \frac{2}{2} = 1 $, $ \frac{3}{2} $, and $ \frac{4}{2} = 2 $ are different!

Q: How do I organize my work when checking multiple slopes?

List all consecutive point pairsCalculate each slope using the formulaCompare results in a clear table or listState conclusion: uniform if all equal, non-uniform if any differ

Rate of Change with Non-Linear Functions

Given a table showing points on the graph of a function, determine whether or not the rate of change is uniform.

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine if the rate of change is uniform?

00:06 It appears that the change in X values is always equal

00:14 However, the change in Y values is not equal

00:18 Therefore, the rate of change is not uniform

00:21 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given a table showing points on the graph of a function, determine whether or not the rate of change is uniform.

Step-by-step solution

To determine if the rate of change is uniform, follow these steps:

Step 1: Calculate the slope between points:

For $(-2, 3)$ to $(0, 5)$ :

m_1 = \frac{5 - 3}{0 + 2} = \frac{2}{2} = 1

For $(0, 5)$ to $(2, 8)$ :

m_2 = \frac{8 - 5}{2 - 0} = \frac{3}{2}

For $(2, 8)$ to $(4, 12)$ :

m_3 = \frac{12 - 8}{4 - 2} = \frac{4}{2} = 2

Step 2: Compare these slopes:
Since $m_1 = 1$ , $m_2 = \frac{3}{2}$ , and $m_3 = 2$ are not equal, the rate of change is not constant.

Therefore, the rate of change is non-uniform.

Final Answer

Non-uniform

Key Points to Remember

Essential concepts to master this topic

Definition: Uniform rate means constant slope between all consecutive points
Technique: Calculate $m = \frac{y_2 - y_1}{x_2 - x_1}$ for each interval separately
Check: Compare all slopes: if any differ, rate is non-uniform ✓

Common Mistakes

Avoid these frequent errors

Calculating only one slope or averaging slopes
Don't just find slope from first to last point ((-2,3) to (4,12)) and conclude it's uniform = missing the changing pattern! This ignores what happens between points. Always calculate slope for every consecutive pair of points and compare them all.

Practice Quiz

Test your knowledge with interactive questions

Look at the graph below and determine whether the function's rate of change is constant or not:

FAQ

Everything you need to know about this question

What's the difference between uniform and non-uniform rate of change?

Uniform rate means the slope is the same between every pair of consecutive points - this creates a straight line. Non-uniform rate means the slopes are different, creating a curved line.

Why can't I just find the slope from the first point to the last point?

That only gives you the average rate of change over the entire interval! To check if it's uniform, you need to see if the rate is constant throughout by checking each step separately.

In this problem, why are the slopes 1, 3/2, and 2 all different?

The y-values increase by 2, then 3, then 4 while x increases by 2 each time. Since the y-changes aren't constant, the slopes $\frac{2}{2} = 1$ , $\frac{3}{2}$ , and $\frac{4}{2} = 2$ are different!

What would the table look like if the rate WAS uniform?

If uniform, all slopes would be equal. For example: (-2,3), (0,5), (2,7), (4,9) would have slopes of 1, 1, 1 - all the same!

How do I organize my work when checking multiple slopes?

List all consecutive point pairs
Calculate each slope using the formula
Compare results in a clear table or list
State conclusion: uniform if all equal, non-uniform if any differ

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