Find the Rate of Change: Analyzing y = -2x + 1

Q: What does the negative sign in -2 mean?

The negative sign means the function is decreasing! As x gets larger, y gets smaller. For every 1 unit increase in x, y decreases by 2 units.

Q: Is the rate of change the same as slope?

Yes! Rate of change and slope are the same thing for linear functions. Both describe how much y changes when x increases by 1.

Q: What if the equation was written differently?

No matter how it's written, always rearrange to slope-intercept form $ y = mx + b $. The coefficient of x will always be your rate of change.

Q: How do I remember which number is the slope?

In $ y = mx + b $, the slope is always the number multiplied by x. The other number (b) stands alone and is the y-intercept.

Q: Can I verify this answer somehow?

Yes! Pick two points on the line and calculate change in y over change in x. For example, when x = 0, y = 1. When x = 1, y = -1. So rate of change = $ \frac{-1-1}{1-0} = -2 $ ✓

Linear Functions with Negative Slopes

Given the linear function:

$y=-2x+1$

What is the rate of change of the function?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:06 Let's find how fast the function changes.

00:09 The rate of change is like the slope. It's how steep the function is.

00:15 We'll use the equation for a straight line to help us.

00:19 Let's compare and determine the slope of the function using this equation.

00:24 And that's how we solve this problem. Nice work!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the linear function:

$y=-2x+1$

What is the rate of change of the function?

Step-by-step solution

To determine the rate of change of the function given by the equation $y = -2x + 1$ , we will follow these steps:

Step 1: Recognize that the equation is written in slope-intercept form, $y = mx + b$ , where $m$ is the slope, or rate of change.
Step 2: Identify the coefficient of $x$ in the given equation. Here, the equation $y = -2x + 1$ reveals that the coefficient of $x$ is $-2$ .

This coefficient $-2$ directly represents the slope $m$ of the function.

Therefore, the rate of change of the function is $m = -2$ .

Final Answer

$m=-2$

Key Points to Remember

Essential concepts to master this topic

Rate of Change: The coefficient of x in y = mx + b
Technique: For y = -2x + 1, coefficient is -2
Check: Negative slope means function decreases as x increases ✓

Common Mistakes

Avoid these frequent errors

Confusing slope with y-intercept
Don't identify the y-intercept (1) as the rate of change = wrong interpretation! The y-intercept tells you where the line crosses the y-axis, not how steep it is. Always identify the coefficient of x as the rate of change.

Practice Quiz

Test your knowledge with interactive questions

For the function in front of you, the slope is?

FAQ

Everything you need to know about this question

What does the negative sign in -2 mean?

The negative sign means the function is decreasing! As x gets larger, y gets smaller. For every 1 unit increase in x, y decreases by 2 units.

Is the rate of change the same as slope?

Yes! Rate of change and slope are the same thing for linear functions. Both describe how much y changes when x increases by 1.

What if the equation was written differently?

No matter how it's written, always rearrange to slope-intercept form $y = mx + b$ . The coefficient of x will always be your rate of change.

How do I remember which number is the slope?

In $y = mx + b$ , the slope is always the number multiplied by x. The other number (b) stands alone and is the y-intercept.

Can I verify this answer somehow?

Yes! Pick two points on the line and calculate change in y over change in x. For example, when x = 0, y = 1. When x = 1, y = -1. So rate of change = $\frac{-1-1}{1-0} = -2$ ✓

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