Finding Decreasing Intervals: Graph Analysis with x=0.65 Reference Line

Q: How do I tell if a function is increasing or decreasing from a graph?

Move your finger from left to right along the curve. If the curve goes down (y-values get smaller), it's decreasing. If it goes up (y-values get larger), it's increasing.

Q: What does the red line at x = 0.65 tell me?

The red line is a reference line to help you analyze the function's behavior. Check what happens to the function on both sides of this line to determine where it increases or decreases.

Q: Can a function be decreasing everywhere?

Yes! Some functions decrease for all values of x. This means no matter which x-value you pick, moving to a larger x always gives you a smaller y-value.

Q: What if the curve looks like it's going up on my screen?

Don't rely on visual direction alone! Focus on the coordinate values. As you move right (increasing x), are the y-coordinates getting larger or smaller? That determines increasing vs decreasing.

Q: How do I write the interval where a function decreases?

If the function decreases everywhere, write $ (-\infty, \infty) $ or 'all real numbers'. If it decreases in specific sections, use interval notation like $ (a, b) $.

Function Behavior with Graph Interpretation

In which interval does the function decrease?

Red line: $x=0.65$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find when the function goes down.

00:09 A function decreases when X goes up, and Y goes down.

00:14 It seems the function is always going down.

00:18 And that's the answer to our question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

In which interval does the function decrease?

Red line: $x=0.65$

Step-by-step solution

Remember that a function is increasing if both the $x$ values and the $y$ values are increasing simultaneously.

A function is decreasing if the $x$ values are increasing while the $y$ values are decreasing simultaneously.

In the graph we can see that the function is decreasing in all domains. In other words, it is decreasing for all $x$ .

Final Answer

All values of $x$

Key Points to Remember

Essential concepts to master this topic

Decreasing Function: As x increases, y values decrease consistently
Visual Analysis: Follow curve from left to right, y drops from top to bottom
Verify: Check any two points: if x₂ > x₁ then f(x₂) < f(x₁) ✓

Common Mistakes

Avoid these frequent errors

Confusing increasing with decreasing based on visual slope
Don't look at whether the curve goes 'up' or 'down' visually = wrong interpretation! The curve's visual direction can be misleading. Always trace from left to right: if y-values get smaller as x gets bigger, the function is decreasing.

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 do I tell if a function is increasing or decreasing from a graph?

Move your finger from left to right along the curve. If the curve goes down (y-values get smaller), it's decreasing. If it goes up (y-values get larger), it's increasing.

What does the red line at x = 0.65 tell me?

The red line is a reference line to help you analyze the function's behavior. Check what happens to the function on both sides of this line to determine where it increases or decreases.

Can a function be decreasing everywhere?

Yes! Some functions decrease for all values of x. This means no matter which x-value you pick, moving to a larger x always gives you a smaller y-value.

What if the curve looks like it's going up on my screen?

Don't rely on visual direction alone! Focus on the coordinate values. As you move right (increasing x), are the y-coordinates getting larger or smaller? That determines increasing vs decreasing.

How do I write the interval where a function decreases?

If the function decreases everywhere, write $(-\infty, \infty)$ or 'all real numbers'. If it decreases in specific sections, use interval notation like $(a, b)$ .

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