Analyzing Linear Graph: Identifying Decreasing Function Behavior

Q: What's the difference between constant and decreasing functions?

A constant function has the same y-value everywhere (horizontal line), while a decreasing function has y-values that get smaller as x increases (slopes downward).

Q: Can a horizontal line ever be considered decreasing?

No! For a function to be decreasing, the y-values must actually decrease as x increases. Since horizontal lines have the same y-value everywhere, they cannot be decreasing.

Q: How do I identify if any line is decreasing?

Look at the line from left to right. If it slopes downward (like going downhill), it's decreasing. If it's flat or slopes upward, it's not decreasing.

Q: What type of function is represented by a horizontal line?

A horizontal line represents a constant function, written as $ f(x) = c $ where c is some constant number. The output never changes regardless of the input.

Q: Why does the slope matter for determining if a function decreases?

The slope tells us how steep the line is. Negative slope means decreasing, positive slope means increasing, and zero slope (horizontal) means constant - not decreasing!

Constant Functions with Horizontal Line Behavior

Is the function in the graph decreasing?

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Is the function in the graph decreasing?

Step-by-step solution

To analyze whether the function in the graph is decreasing, we must understand how the function's behavior is defined by its graph:

Step 1: Examine the graph. The graph presented is a horizontal line.
Step 2: Recognize the properties of a horizontal line. Horizontally aligned lines correspond to constant functions because the $y$ -value remains the same for all $x$ -values.
Step 3: Define the criteria for a function to be decreasing. A function decreases when, as $x$ increases, the value of $f(x)$ decreases.
Step 4: Apply this criterion to the horizontal line. Since the $y$ -value is constant and does not decrease as $x$ moves rightward, the function is not decreasing.

Therefore, the function represented by the graph is not decreasing.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Horizontal lines represent constant functions with same y-value
Technique: Check if y-value changes: constant = not decreasing
Check: Verify slope equals zero for horizontal lines ✓

Common Mistakes

Avoid these frequent errors

Confusing constant with decreasing functions
Don't assume horizontal lines are decreasing because they're not going up = wrong classification! A decreasing function must have y-values that get smaller as x increases. Always check if y-values actually decrease moving left to right.

Practice Quiz

Test your knowledge with interactive questions

Does the function in the graph decrease throughout?

FAQ

Everything you need to know about this question

What's the difference between constant and decreasing functions?

A constant function has the same y-value everywhere (horizontal line), while a decreasing function has y-values that get smaller as x increases (slopes downward).

Can a horizontal line ever be considered decreasing?

No! For a function to be decreasing, the y-values must actually decrease as x increases. Since horizontal lines have the same y-value everywhere, they cannot be decreasing.

How do I identify if any line is decreasing?

Look at the line from left to right. If it slopes downward (like going downhill), it's decreasing. If it's flat or slopes upward, it's not decreasing.

What type of function is represented by a horizontal line?

A horizontal line represents a constant function, written as $f(x) = c$ where c is some constant number. The output never changes regardless of the input.

Why does the slope matter for determining if a function decreases?

The slope tells us how steep the line is. Negative slope means decreasing, positive slope means increasing, and zero slope (horizontal) means constant - not decreasing!

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