Analyze y=-3x+3: Determining if the Linear Function is Increasing or Decreasing

Q: How do I quickly identify if a linear function is increasing or decreasing?

Look at the coefficient of x (the slope). If it's positive, the function is increasing. If it's negative, the function is decreasing. In $ y = -3x + 3 $, the coefficient is -3, so it's decreasing.

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

A decreasing function means as x gets larger, y gets smaller. Think of it like going downhill - as you move right on the graph, the line goes down.

Q: Does the y-intercept affect whether the function is increasing or decreasing?

No! The y-intercept (the +3 in this case) only tells you where the line crosses the y-axis. Only the slope determines if the function increases or decreases.

Q: How can I verify my answer using the graph?

Look at the line from left to right. If it goes upward, it's increasing. If it goes downward, it's decreasing. The graph shows the line going down from left to right, confirming it's decreasing.

Q: What if the slope were positive instead?

If we had $ y = 3x + 3 $ (positive slope), then the function would be increasing. The sign of the slope is the key - positive slopes increase, negative slopes decrease.

Linear Functions with Negative Slopes

Given the following function:

$y=-3x+3$

Is the function increasing or decreasing?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Is the function increasing or decreasing?

00:03 The function equation according to the given data

00:07 The function's slope is negative according to the given data

00:10 When the function's slope is negative, the function is decreasing

00:14 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

Given the following function:

$y=-3x+3$

Is the function increasing or decreasing?

Step-by-step solution

To determine if the function $y = -3x + 3$ is increasing or decreasing, we need to examine the slope of the function.

The function is in the form $y = mx + b$ , where $m$ is the slope.

In this function, the slope $m = -3$ .

The sign of the slope $m$ tells us whether the function is increasing or decreasing:

If $m > 0$ , the function is increasing.
If $m < 0$ , the function is decreasing.

Since $m = -3$ and it is less than zero, the function is decreasing.

Therefore, the function is decreasing.

Final Answer

Decreasing

Key Points to Remember

Essential concepts to master this topic

Slope Rule: Negative slope means function decreases as x increases
Technique: In y = -3x + 3, slope m = -3 < 0
Check: Test two points: when x increases, y decreases ✓

Common Mistakes

Avoid these frequent errors

Confusing slope sign with y-intercept sign
Don't look at the y-intercept (+3) to determine increasing/decreasing = wrong conclusion! The y-intercept only tells you where the line crosses the y-axis, not the direction. Always examine the coefficient of x (the slope) to determine if the function increases or decreases.

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 quickly identify if a linear function is increasing or decreasing?

Look at the coefficient of x (the slope). If it's positive, the function is increasing. If it's negative, the function is decreasing. In $y = -3x + 3$ , the coefficient is -3, so it's decreasing.

What does it mean for a function to be decreasing?

A decreasing function means as x gets larger, y gets smaller. Think of it like going downhill - as you move right on the graph, the line goes down.

Does the y-intercept affect whether the function is increasing or decreasing?

No! The y-intercept (the +3 in this case) only tells you where the line crosses the y-axis. Only the slope determines if the function increases or decreases.

How can I verify my answer using the graph?

Look at the line from left to right. If it goes upward, it's increasing. If it goes downward, it's decreasing. The graph shows the line going down from left to right, confirming it's decreasing.

What if the slope were positive instead?

If we had $y = 3x + 3$ (positive slope), then the function would be increasing. The sign of the slope is the key - positive slopes increase, negative slopes decrease.

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