Analyzing Rate of Change: Determine Uniformity in a Function Table (-1 to 2)

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 equal step in x. This creates a straight line when graphed!

Q: Do I have to check every single pair of consecutive points?

Yes! Even if the first few rates are the same, you must check all consecutive pairs. One different rate makes the entire function non-uniform.

Q: What if I get different rates like 1/4 and 2/8?

Always simplify your fractions! $ \frac{2}{8} = \frac{1}{4} $, so these rates are actually the same. The function would still be uniform.

Q: Can a function have uniform rate of change with negative values?

Absolutely! If all rates are consistently $ -\frac{1}{2} $, for example, that's still uniform. The key is that all rates must be identical, whether positive or negative.

Q: How is this different from finding slope?

Rate of change and slope are the same concept! When checking uniformity in a table, you're essentially verifying that the slope between all consecutive points is constant.

Rate of Change with Function Tables

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?

00:07 It appears that the change in X values is always equal

00:12 It appears that the change in Y values is always equal

00:16 Therefore the rate of change is uniform

00:20 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, we will calculate it between consecutive points and compare them step-by-step:

Step 1: Calculate between $(0, -1)$ and $(4, 0)$ .
$\text{Rate of change} = \frac{0 - (-1)}{4 - 0} = \frac{1}{4}$
Step 2: Calculate between $(4, 0)$ and $(8, 1)$ .
$\text{Rate of change} = \frac{1 - 0}{8 - 4} = \frac{1}{4}$
Step 3: Calculate between $(8, 1)$ and $(12, 2)$ .
$\text{Rate of change} = \frac{2 - 1}{12 - 8} = \frac{1}{4}$
Step 4: Compare the rates from each step.

Since the rate of change is consistently $\frac{1}{4}$ between each pair of points, the rate of change is uniform.

Therefore, the solution to the problem is Uniform.

Final Answer

Uniform

Key Points to Remember

Essential concepts to master this topic

Definition: Rate of change equals change in y divided by change in x
Calculation: Use $\frac{y_2 - y_1}{x_2 - x_1}$ for consecutive points like $\frac{0-(-1)}{4-0} = \frac{1}{4}$
Uniform Check: Calculate rate between all consecutive pairs and verify they're identical ✓

Common Mistakes

Avoid these frequent errors

Calculating rate of change using non-consecutive points
Don't skip points and calculate from (0,-1) to (8,1) = wrong analysis! This misses changes between intermediate points that could show non-uniform rates. 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 equal step in x. This creates a straight line when graphed!

Do I have to check every single pair of consecutive points?

Yes! Even if the first few rates are the same, you must check all consecutive pairs. One different rate makes the entire function non-uniform.

What if I get different rates like 1/4 and 2/8?

Always simplify your fractions! $\frac{2}{8} = \frac{1}{4}$ , so these rates are actually the same. The function would still be uniform.

Can a function have uniform rate of change with negative values?

Absolutely! If all rates are consistently $-\frac{1}{2}$ , for example, that's still uniform. The key is that all rates must be identical, whether positive or negative.

How is this different from finding slope?

Rate of change and slope are the same concept! When checking uniformity in a table, you're essentially verifying that the slope between all consecutive points is constant.

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