Linear Function Analysis: Determining Decreasing Behavior from Graph

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

Look at the line from left to right. If the line goes downward as you move from left to right, it's decreasing. Think of it like walking down a hill!

Q: What's the difference between negative slope and decreasing function?

They're the same thing for linear functions! A negative slope means the function is decreasing. For every step right, you go down.

Q: Can I use any two points to check if it's decreasing?

Yes! Pick any two points on the line. If \( \frac{y_2 - y_1}{x_2 - x_1} , the function is decreasing. Make sure x₂ > x₁.

Linear Functions with Slope Interpretation

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

The graph presented is a straight line. To determine whether the function is decreasing, we need to examine the slope of this line.

The line has a negative slope, as it moves downward from left to right. A function is considered decreasing when its slope is negative.

In formal terms, for a linear function expressed as $y = mx + c$ , if the slope $m$ is negative, the function is decreasing over its entire domain.

From the graph, it's evident that the line has a negative slope, thus indicating that the function is indeed decreasing.

Therefore, the answer to the problem is Yes.

Final Answer

Yes

Key Points to Remember

Essential concepts to master this topic

Rule: A function is decreasing when its slope is negative
Technique: Read the line from left to right: falls down = negative slope
Check: Pick two points and calculate: slope = (y₂-y₁)/(x₂-x₁) < 0 ✓

Common Mistakes

Avoid these frequent errors

Confusing visual direction with mathematical definition
Don't assume a line going up on the graph means increasing = wrong interpretation! Visual 'up' depends on your viewing angle. Always check if y-values decrease as x-values increase from 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

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

Look at the line from left to right. If the line goes downward as you move from left to right, it's decreasing. Think of it like walking down a hill!

What's the difference between negative slope and decreasing function?

They're the same thing for linear functions! A negative slope means the function is decreasing. For every step right, you go down.

Can I use any two points to check if it's decreasing?

Yes! Pick any two points on the line. If $\frac{y_2 - y_1}{x_2 - x_1} < 0$ , the function is decreasing. Make sure x₂ > x₁.

What if the line looks steep? Does that matter?

Steepness doesn't determine if it's increasing or decreasing - only the direction matters. A steep line can be increasing or decreasing depending on which way it slopes.

Is there a difference between 'decreasing' and 'going down'?

In math, 'decreasing' has a specific meaning: as x increases, y decreases. Always think left to right when determining if a function is increasing or decreasing.

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