Analyzing Linear Function: Determining Decreasing Behavior from Graph

Q: How can I tell if a line is decreasing just by looking at it?

Look at the direction of the slope! If the line goes from upper-left to lower-right, it's decreasing. Think of it like going downhill as you move from left to right.

Q: What does it mean mathematically for a function to be decreasing?

For any two x-values where $ x_1 , we must have \( f(x_1) > f(x_2) $. This means smaller x-values give larger y-values.

Q: Can a function be both increasing and decreasing?

Not at the same time! A function can be increasing on some intervals and decreasing on others, but linear functions like this one have the same behavior everywhere.

Q: Is there a difference between decreasing and negative slope?

They're related but different! A negative slope is the number (like -2), while decreasing describes the behavior. All lines with negative slopes are decreasing functions.

Q: How do I check my answer without just looking at the graph?

Pick any two points on the line. If the point with the smaller x-coordinate has a larger y-coordinate, then the function is decreasing. This confirms what you see visually!

Linear Functions with Slope Direction

Is the function shown in the graph below 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 shown in the graph below decreasing?

Step-by-step solution

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

Step 1: Visually inspect the graph to see if it is consistently sloping downward.
Step 2: Apply the definition of a decreasing function.

Now, let's work through each step:
Step 1: Observing the graph, the function's graph is a line moving from the top left to the bottom right. This indicates it slopes downward as we move from left to right across the $x$ -axis.
Step 2: According to the definition of a decreasing function, for any $x_1 < x_2$ , it must hold true that $f(x_1) > f(x_2)$ . Since the graph shows a line moving downward, this condition is satisfied throughout its domain.

Therefore, the function represented by the graph is indeed decreasing.

The final answer is Yes.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Definition: Decreasing function means as x increases, y decreases
Visual Check: Graph slopes downward from left to right
Verification: Pick two points: left point higher than right point ✓

Common Mistakes

Avoid these frequent errors

Confusing increasing vs decreasing direction
Don't look at the graph from right to left = wrong conclusion! This reverses the actual behavior of the function. Always read graphs from left to right to determine if y-values are getting smaller (decreasing) or larger (increasing).

Practice Quiz

Test your knowledge with interactive questions

Is the function in the graph decreasing?

FAQ

Everything you need to know about this question

How can I tell if a line is decreasing just by looking at it?

Look at the direction of the slope! If the line goes from upper-left to lower-right, it's decreasing. Think of it like going downhill as you move from left to right.

What does it mean mathematically for a function to be decreasing?

For any two x-values where $x_1 < x_2$ , we must have $f(x_1) > f(x_2)$ . This means smaller x-values give larger y-values.

Can a function be both increasing and decreasing?

Not at the same time! A function can be increasing on some intervals and decreasing on others, but linear functions like this one have the same behavior everywhere.

Is there a difference between decreasing and negative slope?

They're related but different! A negative slope is the number (like -2), while decreasing describes the behavior. All lines with negative slopes are decreasing functions.

How do I check my answer without just looking at the graph?

Pick any two points on the line. If the point with the smaller x-coordinate has a larger y-coordinate, then the function is decreasing. This confirms what you see visually!

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