Analyzing Rate of Change: Compare Points from (-2,1) to (1,7) in Function Table

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 each step is exactly the same height!

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 it's non-uniform. Don't skip any pairs!

Q: What if I get the same rate but it's negative?

That's still uniform! A negative rate just means the function is decreasing at a constant rate instead of increasing. The sign doesn't affect uniformity.

Q: How can I tell if my calculations are right?

Double-check by using the formula carefully: $ \frac{\text{new y} - \text{old y}}{\text{new x} - \text{old x}} $. Make sure you subtract in the same order for both numerator and denominator!

Q: What does it mean if the rates are different?

Different rates mean non-uniform change! This indicates the function is curving (like a parabola) rather than forming a straight line.

Rate of Change with Consecutive 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?

00:07 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 edge of the function, determine whether the rate of change is uniform or not.

Step-by-step solution

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

Step 1: Calculate the rate of change between each consecutive pair of points.
Step 2: Compare these rates to check for uniformity.

Now, let's work through each step:
Step 1: Calculate the rate of change for each consecutive pair of points:
- Between $(-2, 1)$ and $(-1, 3)$ :
$\frac{3 - 1}{-1 - (-2)} = \frac{2}{1} = 2$
- Between $(-1, 3)$ and $(0, 5)$ :
$\frac{5 - 3}{0 - (-1)} = \frac{2}{1} = 2$
- Between $(0, 5)$ and $(1, 7)$ :
$\frac{7 - 5}{1 - 0} = \frac{2}{1} = 2$

Step 2: Compare the calculated rates of change.
We observe that the rate of change is constantly $2$ for each pair of points.

Therefore, the solution to the problem is that the rate of change is Uniform.

Final Answer

Uniform

Key Points to Remember

Essential concepts to master this topic

Formula: Rate of change equals $\frac{y_2 - y_1}{x_2 - x_1}$ between two points
Technique: Calculate $\frac{3-1}{-1-(-2)} = \frac{2}{1} = 2$ for each consecutive pair
Check: If all rates equal the same value, then uniform ✓

Common Mistakes

Avoid these frequent errors

Calculating rate of change incorrectly by mixing up coordinates
Don't use $\frac{x_2 - x_1}{y_2 - y_1}$ = wrong formula! This gives the reciprocal of the actual rate and leads to completely incorrect analysis. Always use $\frac{\text{change in y}}{\text{change in x}}$ in that exact order.

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 each step is exactly the same height!

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 it's non-uniform. Don't skip any pairs!

What if I get the same rate but it's negative?

That's still uniform! A negative rate just means the function is decreasing at a constant rate instead of increasing. The sign doesn't affect uniformity.

How can I tell if my calculations are right?

Double-check by using the formula carefully: $\frac{\text{new y} - \text{old y}}{\text{new x} - \text{old x}}$ . Make sure you subtract in the same order for both numerator and denominator!

What does it mean if the rates are different?

Different rates mean non-uniform change! This indicates the function is curving (like a parabola) rather than forming a straight line.

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