Analyzing Rate of Change: Function Table with X-Values (3,5,7,9)

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

Uniform means all rates are the same (like 2, 2, 2). Non-uniform means at least one rate is different (like $ \frac{1}{2} $, 2, 2).

Q: Can I use any two points to find the rate?

For determining uniformity, you must use consecutive pairs only. Using random points like (3,-2) and (9,7) won't show the true pattern of change.

Q: What if I get the same rate for some pairs but not others?

That still means non-uniform! All rates must be identical for uniform rate of change. Even one different rate makes it non-uniform.

Q: How do I organize my calculations neatly?

List each consecutive pair: (3,-2) to (5,-1)Show the formula: $ \frac{-1-(-2)}{5-3} $Simplify: $ \frac{1}{2} $Repeat for all pairs

Rate of Change with Variable Differences

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

00:19 Let's check if the rate of change is equal everywhere.

00:23 Notice, the change in X values is always the same. That's good!

00:30 But, look carefully. The change in Y values is not the same.

00:34 So, the rate of change is not constant.

00:38 And that's how we solve this problem. Great job!

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 whether the rate of change is uniform, we will calculate the rate of change between each pair of consecutive points given.

Step 1: Calculate the rate of change between consecutive points.

Calculate between $(3, -2)$ and $(5, -1)$ :

\frac{-1 - (-2)}{5 - 3} = \frac{1}{2}

Calculate between $(5, -1)$ and $(7, 3)$ :

\frac{3 - (-1)}{7 - 5} = \frac{4}{2} = 2

Calculate between $(7, 3)$ and $(9, 7)$ :

\frac{7 - 3}{9 - 7} = \frac{4}{2} = 2

Step 2: Analyze the calculated rates of change.

We observe that the calculated rates of change are $\frac{1}{2}$ , $2$ , and $2$ . Since the first calculated rate of change is different from the others, the rate of change between the points is not consistent.

Therefore, the rate of change is non-uniform.

Final Answer

Non-uniform

Key Points to Remember

Essential concepts to master this topic

Definition: Rate of change equals rise over run between points
Calculation: Use $\frac{y_2 - y_1}{x_2 - x_1}$ for each consecutive pair
Check: Compare all rates: $\frac{1}{2}$ , 2, 2 are different = non-uniform ✓

Common Mistakes

Avoid these frequent errors

Calculating only one rate of change
Don't calculate just between first and last points = you miss the pattern! This hides that rates change differently between consecutive intervals. Always calculate the 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's the difference between uniform and non-uniform rate of change?

Uniform means all rates are the same (like 2, 2, 2). Non-uniform means at least one rate is different (like $\frac{1}{2}$ , 2, 2).

Why do I need to check all consecutive pairs?

Each pair shows how the function behaves in that specific interval. If you skip pairs, you might miss where the rate changes and get the wrong answer!

Can I use any two points to find the rate?

For determining uniformity, you must use consecutive pairs only. Using random points like (3,-2) and (9,7) won't show the true pattern of change.

What if I get the same rate for some pairs but not others?

That still means non-uniform! All rates must be identical for uniform rate of change. Even one different rate makes it non-uniform.

How do I organize my calculations neatly?

List each consecutive pair: (3,-2) to (5,-1)
Show the formula: $\frac{-1-(-2)}{5-3}$
Simplify: $\frac{1}{2}$
Repeat for all pairs

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