Given the following graph, determine whether the rate of change is uniform or not–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010111111–3–3–3–2–2–2–1–1–1111222333444555000

The graph does not represents a function

Graph Analysis: Determining Uniform Rate of Change in a Semi-Circular Function

Q: How can I tell if a rate of change is uniform just by looking?

Look for straightness! If the graph is a perfectly straight line, the rate of change is uniform. Any curve, bend, or change in direction means the rate varies.

Q: What makes this semi-circle have non-uniform rate of change?

The curvature is key! At the beginning and end, the curve is nearly flat (small rate of change), but in the middle it's steep (large rate of change). This variation makes it non-uniform.

Q: Could a function be continuous but still have non-uniform rate of change?

Absolutely! Continuity means no breaks in the graph, while uniform rate of change means constant slope. A smooth curve like this semi-circle is continuous but has varying slopes.

Q: How do I calculate the actual rate of change for curves?

For curves, you calculate the average rate of change between two points using $ \frac{y_2 - y_1}{x_2 - x_1} $. Different intervals will give different values, proving it's non-uniform.

Q: Why is this called a 'function' if it looks like half a circle?

It's a function because it passes the vertical line test - every x-value has exactly one corresponding y-value. The upper half of a circle is indeed a valid function!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is the rate of change of the function uniform?

00:03 We want to check if the value differences in X and Y are constant

00:07 For this, we'll take several points on the graph and observe the rate of change

00:33 Let's calculate the differences between X values

00:36 The differences in X values are equal

00:40 Let's calculate the differences between Y values

00:43 We can see that the differences between Y values are not equal

00:46 Therefore, the rate of change is not uniform

00:50 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 the following graph, determine whether the rate of change is uniform or not

2

Step-by-step solution

The problem asks us to determine if the rate of change in the graph is uniform or not. To do this, we need to examine the graph closely to see whether it is linear.

If a graph is linear, it means it is a straight line, indicating a constant (uniform) rate of change. The slope of a straight line does not change, meaning that for every unit increase in $x$ there is a proportional and consistent change in $y$ .

In contrast, if a graph curves or the line is not straight, the rate of change would not be uniform. This is because a curve indicates that the amount $y$ changes for each unit change in $x$ is not constant.

By analyzing the given graph, we can see that it is a non-linear function with a visible curve. Since the line is not straight (it appears as a curved line in the graph), the rate of change of the function is not constant across its range.

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

Consequently, the correct choice, corresponding to a non-uniform rate of change in the graph, is:

Non-uniform

3

Final Answer

Non-uniform

Key Points to Remember

Essential concepts to master this topic

Linear vs Non-Linear: Straight lines have uniform rate of change, curves do not
Visual Analysis: Semi-circle from (0,0) to (8,0) shows clear curvature
Verification: Calculate slope between different intervals: varies greatly ✓

Common Mistakes

Avoid these frequent errors

Confusing function type with rate of change uniformity
Don't assume all functions have uniform rates just because they're continuous = wrong analysis! A smooth curve can still have varying rates of change at different points. Always examine if the graph is a straight line to determine uniform rate of change.

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

How can I tell if a rate of change is uniform just by looking?

+

Look for straightness! If the graph is a perfectly straight line, the rate of change is uniform. Any curve, bend, or change in direction means the rate varies.

What makes this semi-circle have non-uniform rate of change?

+

The curvature is key! At the beginning and end, the curve is nearly flat (small rate of change), but in the middle it's steep (large rate of change). This variation makes it non-uniform.

Could a function be continuous but still have non-uniform rate of change?

+

Absolutely! Continuity means no breaks in the graph, while uniform rate of change means constant slope. A smooth curve like this semi-circle is continuous but has varying slopes.

How do I calculate the actual rate of change for curves?

+

For curves, you calculate the average rate of change between two points using $\frac{y_2 - y_1}{x_2 - x_1}$ . Different intervals will give different values, proving it's non-uniform.

Why is this called a 'function' if it looks like half a circle?

+

It's a function because it passes the vertical line test - every x-value has exactly one corresponding y-value. The upper half of a circle is indeed a valid function!

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Graph Analysis: Determining Uniform Rate of Change in a Semi-Circular Function

Rate of Change with Non-Linear Functions

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