Analyzing Rate of Change: Function Table with X-Y Coordinates (1,2) to (4,7)

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. It's like climbing stairs where every step is the same height!

Q: Why do I need to check between consecutive points instead of just the endpoints?

Checking only endpoints gives you the average rate, not whether the change is uniform throughout. You need to verify that every single interval has the same rate of change.

Q: How can I tell if a function has uniform rate of change just by looking at the table?

Look for a constant difference in y-values when x-values increase by 1. In this problem: y increases by 1, then 1, then 3 - not constant, so non-uniform!

Q: What type of function has uniform rate of change?

Only linear functions (straight lines) have uniform rates of change. If you plotted these points, you'd see the line bends at (3,4), showing it's not linear.

Q: Is there a quick way to spot non-uniform rate without calculating?

Yes! If the y-values don't increase by the same amount for each step in x, the rate is non-uniform. Here: +1, +1, +3 shows the pattern breaks.

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

00:13 However, the change in Y values is not equal

00:16 Therefore, the rate of change is not uniform

00:19 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

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

2

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Calculate the rate of change between the first pair of points $(1, 2)$ and $(2, 3)$ .
Step 2: Calculate the rate of change between the second pair of points $(2, 3)$ and $(3, 4)$ .
Step 3: Calculate the rate of change between the third pair of points $(3, 4)$ and $(4, 7)$ .
Step 4: Compare the calculated rates to determine uniformity.

Step 1: Calculate the slope between $(1, 2)$ and $(2, 3)$
$\text{Slope} = \frac{3 - 2}{2 - 1} = 1$

Step 2: Calculate the slope between $(2, 3)$ and $(3, 4)$
$\text{Slope} = \frac{4 - 3}{3 - 2} = 1$

Step 3: Calculate the slope between $(3, 4)$ and $(4, 7)$
$\text{Slope} = \frac{7 - 4}{4 - 3} = 3$

Step 4: Compare the slopes:
The slopes between the first two pairs of points are equal to 1, while the slope between the last pair of points is 3. Since these slopes are not equal, the rate of change is not uniform.

Therefore, the solution to the problem is that the rate of change is non-uniform.

3

Final Answer

Non-uniform

Key Points to Remember

Essential concepts to master this topic

Rate Formula: Use slope formula (y₂ - y₁) ÷ (x₂ - x₁) between consecutive points
Technique: Calculate each interval: 1→1, 1→1, then 3→1, showing non-uniform change
Check: Compare all rates; if any differ, function has non-uniform rate ✓

Common Mistakes

Avoid these frequent errors

Calculating only the overall rate from first to last point
Don't just calculate from (1,2) to (4,7) getting rate = 5/3 and call it uniform! This ignores what happens between points and misses the actual pattern. Always calculate the rate between each consecutive pair of points to check uniformity.

Practice Quiz

Test your knowledge with interactive questions

Given the following graph, determine whether function is constant

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. It's like climbing stairs where every step is the same height!

Why do I need to check between consecutive points instead of just the endpoints?

+

Checking only endpoints gives you the average rate, not whether the change is uniform throughout. You need to verify that every single interval has the same rate of change.

How can I tell if a function has uniform rate of change just by looking at the table?

+

Look for a constant difference in y-values when x-values increase by 1. In this problem: y increases by 1, then 1, then 3 - not constant, so non-uniform!

What type of function has uniform rate of change?

+

Only linear functions (straight lines) have uniform rates of change. If you plotted these points, you'd see the line bends at (3,4), showing it's not linear.

Is there a quick way to spot non-uniform rate without calculating?

+

Yes! If the y-values don't increase by the same amount for each step in x, the rate is non-uniform. Here: +1, +1, +3 shows the pattern breaks.

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Analyzing Rate of Change: Function Table with X-Y Coordinates (1,2) to (4,7)

Rate of Change with Consecutive Points

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