Analyzing Rate of Change: X-Values 8-11 with Corresponding Y-Values

Q: What does 'uniform rate of change' actually mean?

A uniform rate of change means the function increases (or decreases) by the same amount for every unit increase in x. It's like climbing stairs where each step is exactly the same height!

Q: Why do I need to check ALL consecutive pairs?

You need to verify every single step is the same! Even if the first two rates match, the third one might be different. Think of it like checking that all stairs are the same height - you can't skip any!

Q: What if I get different rates for different pairs?

Then the rate of change is non-uniform! This means the function doesn't increase steadily - it might speed up or slow down at different intervals.

Q: Can the rate of change be negative and still uniform?

Absolutely! If you get the same negative number for all consecutive pairs (like -3, -3, -3), then it's still uniform. The function is just decreasing at a steady rate.

Q: What does this tell me about the type of function?

A uniform rate of change means you have a linear function! The points form a straight line. Non-uniform rates suggest curves, parabolas, or other non-linear relationships.

Rate of Change with Consecutive Data Points

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

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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?

00:05 It appears that the change in X values is always equal

00:09 It appears that the change in Y values is always equal

00:12 Therefore, the rate of change is uniform

00:15 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 edge of the function, determine whether the rate of change is uniform or not.

Step-by-step solution

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

Step 1: Calculate the rate of change between consecutive points.
Step 2: Compare these rates to see if they are consistent.

Let's work through each step:

Step 1: Calculate the rate of change.

For points $(8, 2)$ and $(9, 4)$ :

\text{Rate of change} = \frac{4 - 2}{9 - 8} = \frac{2}{1} = 2

For points $(9, 4)$ and $(10, 6)$ :

\text{Rate of change} = \frac{6 - 4}{10 - 9} = \frac{2}{1} = 2

For points $(10, 6)$ and $(11, 8)$ :

\text{Rate of change} = \frac{8 - 6}{11 - 10} = \frac{2}{1} = 2

Step 2: Comparing the rates of change, we find they are all equal to 2, indicating uniformity.

Therefore, the rate of change is Uniform.

Final Answer

Uniform

Key Points to Remember

Essential concepts to master this topic

Rate Formula: Calculate using $\frac{\text{change in y}}{\text{change in x}}$ between consecutive points
Technique: Find $\frac{4-2}{9-8} = \frac{2}{1} = 2$ for each pair systematically
Check: Compare all calculated rates: if identical, then uniform ✓

Common Mistakes

Avoid these frequent errors

Skipping points when calculating rates
Don't calculate rate from (8,2) to (10,6) directly = rate of 2, but this misses checking uniformity! This skips the middle calculation and can hide non-uniform sections. Always calculate rate between every consecutive pair of points.

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 does 'uniform rate of change' actually mean?

A uniform rate of change means the function increases (or decreases) by the same amount for every unit increase in x. It's like climbing stairs where each step is exactly the same height!

Why do I need to check ALL consecutive pairs?

You need to verify every single step is the same! Even if the first two rates match, the third one might be different. Think of it like checking that all stairs are the same height - you can't skip any!

What if I get different rates for different pairs?

Then the rate of change is non-uniform! This means the function doesn't increase steadily - it might speed up or slow down at different intervals.

Can the rate of change be negative and still uniform?

Absolutely! If you get the same negative number for all consecutive pairs (like -3, -3, -3), then it's still uniform. The function is just decreasing at a steady rate.

How do I organize my work to avoid mistakes?

List all consecutive point pairs: (8,2)→(9,4), (9,4)→(10,6), (10,6)→(11,8)
Calculate each rate separately and clearly
Compare all results at the end

What does this tell me about the type of function?

A uniform rate of change means you have a linear function! The points form a straight line. Non-uniform rates suggest curves, parabolas, or other non-linear relationships.

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