Given the following graph, determine whether the rate of change is uniform or not–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888–3–3–3–2–2–2–1–1–1111222333444000

The graph does not represents a function

Analyzing Rate of Change: Determining Uniformity in a Horizontal Line Graph

Q: How can zero be considered a uniform rate of change?

Zero is absolutely uniform! A rate of change of zero means the y-value stays exactly the same no matter how x changes. This is perfectly consistent and uniform.

Q: What's the difference between uniform and non-uniform rates?

A uniform rate means the slope stays constant (like straight lines). A non-uniform rate means the slope changes as you move along the curve (like parabolas or circles).

Q: Why does a horizontal line represent a function?

A horizontal line passes the vertical line test - every x-value has exactly one y-value. It's a valid function, just one where the output never changes!

Q: How do I calculate the rate of change for any two points?

Use the formula: $ \text{Rate} = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1} $. For horizontal lines, the numerator is always zero.

Q: Can I have a uniform rate that isn't zero?

Absolutely! Any straight line has a uniform rate of change. A line with slope 2 has a uniform rate of +2, while a line with slope -3 has a uniform rate of -3.

Rate of Change with Horizontal Lines

Given the following graph, 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 Is the rate of change uniform?

00:05 Let's take some points on the graph

00:16 We can see that each time the change is equal

00:23 Therefore the graph has a uniform rate of change

00:27 And this is the solution to the question

Step-by-step written solution

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Understand the problem

Given the following graph, determine whether the rate of change is uniform or not

Step-by-step solution

The problem requires us to determine whether the rate of change in a given graph is uniform.

A uniform rate of change corresponds to a constant slope, which is characteristic of a linear graph. First, we'll examine the graphical representation.

Upon observing the graph, we see that it displays a straight horizontal line. A horizontal line on a graph indicates that for any two points $(x_1, y_1)$ and $(x_2, y_2)$ , the difference in $y$ -values is zero, i.e., $y_2 - y_1 = 0$ . This implies that the slope, given by the formula $\frac{y_2 - y_1}{x_2 - x_1}$ , is zero and remains constant as we move along the line.

Because the line is horizontal and does not change its slope throughout, the rate of change is indeed uniform across the entire graph.

Therefore, the rate of change is uniform.

Final Answer

Uniform

Key Points to Remember

Essential concepts to master this topic

Horizontal Line Rule: A horizontal line has zero slope and constant rate of change
Rate of Change Formula: $\frac{y_2 - y_1}{x_2 - x_1} = \frac{0}{any\ value} = 0$
Verification: Pick any two points: same y-value confirms uniform rate ✓

Common Mistakes

Avoid these frequent errors

Confusing horizontal with vertical lines
Don't think a horizontal line means "no rate of change" = non-uniform! A horizontal line actually has a perfectly uniform rate of change of zero. Always remember: horizontal = uniform zero rate, vertical = undefined rate.

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

How can zero be considered a uniform rate of change?

Zero is absolutely uniform! A rate of change of zero means the y-value stays exactly the same no matter how x changes. This is perfectly consistent and uniform.

What's the difference between uniform and non-uniform rates?

A uniform rate means the slope stays constant (like straight lines). A non-uniform rate means the slope changes as you move along the curve (like parabolas or circles).

Why does a horizontal line represent a function?

A horizontal line passes the vertical line test - every x-value has exactly one y-value. It's a valid function, just one where the output never changes!

How do I calculate the rate of change for any two points?

Use the formula: $\text{Rate} = \frac{\text{change in y}}{\text{change in x}} = \frac{y_2 - y_1}{x_2 - x_1}$ . For horizontal lines, the numerator is always zero.

Can I have a uniform rate that isn't zero?

Absolutely! Any straight line has a uniform rate of change. A line with slope 2 has a uniform rate of +2, while a line with slope -3 has a uniform rate of -3.

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