Analyzing Rate of Change: X-Y Table Analysis from 2 to 8

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-values. This creates a straight line graph!

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

Yes! You must calculate the rate between all consecutive pairs. If even one pair has a different rate, then the overall rate is non-uniform.

Q: What if I get different rates like 1.5, 2, and 1.5?

Then the rate of change is non-uniform! All rates must be exactly equal for the rate to be considered uniform.

Q: How can I double-check my slope calculations?

Use the points in your table: $ \frac{\text{change in y}}{\text{change in x}} $. For (2,3) to (4,6): $ \frac{6-3}{4-2} = \frac{3}{2} = 1.5 $. Always verify each step!

Rate of Change with Tabular Data

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?

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

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

00:15 Therefore, the rate of change is uniform

00:18 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 solve this problem, we'll follow these steps:

Step 1: Calculate the rate of change between each pair of consecutive points.
Step 2: Compare the computed rates to determine if they are equal.
Step 3: Conclude whether the rate of change is uniform or not.

Now, let's work through each step:
Step 1: Calculate the rate of change between each pair of consecutive points:
- Between $(2, 3)$ and $(4, 6)$ :
$\text{Slope} = \frac{6 - 3}{4 - 2} = \frac{3}{2} = 1.5$
- Between $(4, 6)$ and $(6, 9)$ :
$\text{Slope} = \frac{9 - 6}{6 - 4} = \frac{3}{2} = 1.5$
- Between $(6, 9)$ and $(8, 12)$ :
$\text{Slope} = \frac{12 - 9}{8 - 6} = \frac{3}{2} = 1.5$

Step 2: Compare the computed rates:
- In all cases, the rate of change is $1.5$ .

Step 3: Conclude that the rate of change is uniform across the expressed intervals.

Therefore, the rate of change for the given points is uniform.

Final Answer

Uniform

Key Points to Remember

Essential concepts to master this topic

Formula: Rate of change = $\frac{y_2 - y_1}{x_2 - x_1}$ between consecutive points
Technique: Calculate $\frac{6-3}{4-2} = \frac{3}{2} = 1.5$ for each interval
Check: All rates equal means uniform: 1.5 = 1.5 = 1.5 ✓

Common Mistakes

Avoid these frequent errors

Calculating rates incorrectly or skipping intervals
Don't subtract x-values in wrong order like (2-4) instead of (4-2) = negative slope when it should be positive! This gives opposite direction. Always use consistent order: later point minus earlier point for both x and y values.

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-values. This creates a straight line graph!

Do I need to check every single pair of points?

Yes! You must calculate the rate between all consecutive pairs. If even one pair has a different rate, then the overall rate is non-uniform.

What if I get different rates like 1.5, 2, and 1.5?

Then the rate of change is non-uniform! All rates must be exactly equal for the rate to be considered uniform.

Can the rate of change be negative and still uniform?

Absolutely! If all your calculations give the same negative value (like -2, -2, -2), then you have a uniform negative rate. The sign doesn't matter for uniformity.

How can I double-check my slope calculations?

Use the points in your table: $\frac{\text{change in y}}{\text{change in x}}$ . For (2,3) to (4,6): $\frac{6-3}{4-2} = \frac{3}{2} = 1.5$ . Always verify each step!

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