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

What is the solution to the following inequality?

$$ 10x-4\leq-3x-8 $$

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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