Look at the graph below and determine whether the function's rate of change is constant 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.

Analyzing Rate of Change: Determine if a Curved Function has Constant Slope

Q: How can I tell if a function has constant rate of change just by looking?

Look for straight lines! Only linear functions (straight lines) have constant rate of change. If the graph curves, bends, or changes direction at all, the rate of change is not constant.

Q: What does 'rate of change' actually mean?

Rate of change is how much the y-value changes compared to the x-value - it's the slope! For straight lines, this slope stays the same everywhere. For curves, the slope changes as you move along the line.

Q: Why doesn't this curved function have constant slope?

Because it's curved! At the steep parts (left side), the function rises quickly. At the flatter parts (right side), it rises more slowly. This changing steepness means the slope varies.

Q: Can I calculate the exact rate of change for curved functions?

For curves, you can find the average rate of change between two points using $ \frac{\text{change in y}}{\text{change in x}} $, but it won't be the same for different intervals!

Q: What would a graph with constant rate of change look like?

It would be a perfectly straight line! Whether going up, down, or horizontal, straight lines have the same slope everywhere, which means constant rate of change.

Curved Function with Slope Analysis

Look at the graph below and determine whether the function's rate of change is constant or not:

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Step-by-step video solution

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Step-by-step written solution

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Understand the problem

Look at the graph below and determine whether the function's rate of change is constant or not:

Step-by-step solution

First we need to remember that if the function is not a straight line, its rate of change is not constant.

The rate of change is not uniform since the function is not a straight line.

Final Answer

Not constant

Key Points to Remember

Essential concepts to master this topic

Rule: Only straight lines have constant rate of change
Technique: Curved paths show varying steepness at different points
Check: Compare slopes between different intervals to confirm variation ✓

Common Mistakes

Avoid these frequent errors

Assuming any smooth curve has constant slope
Don't think smooth curves mean constant rate = wrong analysis! Smoothness doesn't equal constant slope - curves change direction and steepness continuously. Always remember only straight lines have constant 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 function has constant rate of change just by looking?

Look for straight lines! Only linear functions (straight lines) have constant rate of change. If the graph curves, bends, or changes direction at all, the rate of change is not constant.

What does 'rate of change' actually mean?

Rate of change is how much the y-value changes compared to the x-value - it's the slope! For straight lines, this slope stays the same everywhere. For curves, the slope changes as you move along the line.

Why doesn't this curved function have constant slope?

Because it's curved! At the steep parts (left side), the function rises quickly. At the flatter parts (right side), it rises more slowly. This changing steepness means the slope varies.

Can I calculate the exact rate of change for curved functions?

For curves, you can find the average rate of change between two points using $\frac{\text{change in y}}{\text{change in x}}$ , but it won't be the same for different intervals!

What would a graph with constant rate of change look like?

It would be a perfectly straight line! Whether going up, down, or horizontal, straight lines have the same slope everywhere, which means constant rate of change.

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