Analyzing f(x): Determine the Linear Function's Increasing or Decreasing Behavior

Q: How do I tell if a line is increasing or decreasing just by looking?

Follow the line from left to right (increasing x direction). If the line goes upward, it's increasing. If it goes downward, it's decreasing. If it stays level, it's constant.

Q: Can I use the slope to determine this?

Absolutely! If the slope is positive, the function is increasing. If the slope is negative, it's decreasing. If the slope is zero, it's constant.

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

A decreasing function means that as the input (x) gets larger, the output $ f(x) $ gets smaller. It's like going downhill - the further you walk forward, the lower your elevation becomes.

Linear Function Behavior with Graph Analysis

Here is a graph of a function. $f(x)$

Is the function increasing, decreasing or constant?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Determine whether the function is increasing, decreasing, or constant?

00:03 For this, let's take several points on the graph and observe the rate of change

00:40 We can see that the Y values are decreasing, therefore the function is decreasing

00:47 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Here is a graph of a function. $f(x)$

Is the function increasing, decreasing or constant?

Step-by-step solution

To solve this problem, we'll analyze the graph of the function:

Step 1: Carefully observe the given graph of $f(x)$ .
Step 2: Evaluate the overall trend of the graph.
Step 3: Determine if the function is increasing, decreasing, or constant based on the slope.

Now, let's go through the solution:
Step 1: Observing the graph, we see that the curve starts at a higher coordinate on the y-axis and moves downwards as it progresses from left to right.
Step 2: This indicates a negative slope, as the y-values decrease while the x-values increase.
Step 3: Reasoning from the previous observations, it is clear that as $x$ increases, the function's value $f(x)$ decreases.

Therefore, the correct conclusion is that the function $f(x)$ is decreasing.

Final Answer

Decreasing

Key Points to Remember

Essential concepts to master this topic

Visual Analysis: Examine the line's direction from left to right
Technique: If line goes down as x increases, slope is negative = decreasing
Check: Pick two points: higher x should give lower $f(x)$ value ✓

Common Mistakes

Avoid these frequent errors

Confusing visual direction with mathematical behavior
Don't look at where the line "goes up" on the paper = wrong interpretation! A line can appear to go "up" on the graph but still be decreasing mathematically. Always focus on: as x increases (moving right), does f(x) increase or decrease?

Practice Quiz

Test your knowledge with interactive questions

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

FAQ

Everything you need to know about this question

How do I tell if a line is increasing or decreasing just by looking?

Follow the line from left to right (increasing x direction). If the line goes upward, it's increasing. If it goes downward, it's decreasing. If it stays level, it's constant.

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

The physical direction on your screen doesn't matter! What matters is the mathematical relationship. As x-values get bigger (moving right), do the y-values get bigger or smaller?

Can I use the slope to determine this?

Absolutely! If the slope is positive, the function is increasing. If the slope is negative, it's decreasing. If the slope is zero, it's constant.

What does it mean for a function to be decreasing?

A decreasing function means that as the input (x) gets larger, the output $f(x)$ gets smaller. It's like going downhill - the further you walk forward, the lower your elevation becomes.

How can I check my answer without looking at the graph again?

Pick any two points on the line where the first has a smaller x-value than the second. If the first point has a larger y-value than the second point, the function is decreasing!

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