Find the Rate of Change: Analyzing y = -6x Linear Function

Q: Why is the rate of change negative in this problem?

The rate of change is negative (-6) because as x increases, y decreases. This means the line slopes downward from left to right - for every 1 unit x increases, y drops by 6 units.

Q: What's the difference between slope and rate of change?

They're the same thing! Rate of change is just another way to say slope. Both tell you how much y changes when x changes by 1 unit.

Q: How do I find rate of change if the equation looks different?

First, rearrange the equation to slope-intercept form $ y = mx + b $. Then the coefficient of x is your rate of change, no matter how the original equation was written!

Q: Can rate of change be a fraction or decimal?

Absolutely! Rate of change can be any real number - positive, negative, whole numbers, fractions, or decimals. In this case, it's the whole number -6.

Slope Identification with Direct Linear Functions

Given the linear function:

$y=-6x$

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

00:00 Find the rate of change of the function

00:03 The rate of change is the slope of the function (M)

00:07 We will use the line equation

00:10 We will compare and find the slope of the function using the line equation

00:15 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 linear function:

$y=-6x$

What is the rate of change of the function?

Step-by-step solution

To solve this problem, let's follow these steps:

Step 1: Identify the form of the given function
Step 2: Compare the equation with the standard slope-intercept form
Step 3: Extract the value of the slope from the equation

Now, let's work through each step:

Step 1: The given linear function is $y = -6x$ . This is presented in the form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Step 2: Comparing $y = -6x$ with $y = mx + b$ , we see that the equation lacks a constant term, indicating $b = 0$ . The slope $m$ is the coefficient of $x$ .

Step 3: The coefficient of $x$ is $-6$ , so the slope $m$ is $-6$ . Thus, the rate of change of the function is $-6$ .

Therefore, the solution to the problem is $m = -6$ .

Final Answer

$m=-6$

Key Points to Remember

Essential concepts to master this topic

Rule: In $y = mx + b$ form, m equals the rate of change
Technique: Coefficient of x in $y = -6x$ gives slope = -6
Check: Verify rate of change: when x increases by 1, y decreases by 6 ✓

Common Mistakes

Avoid these frequent errors

Confusing slope with y-intercept
Don't look for the constant term when finding slope = wrong answer! The y-intercept (b) doesn't affect rate of change. Always identify the coefficient of x as your slope value.

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

Why is the rate of change negative in this problem?

The rate of change is negative (-6) because as x increases, y decreases. This means the line slopes downward from left to right - for every 1 unit x increases, y drops by 6 units.

What's the difference between slope and rate of change?

They're the same thing! Rate of change is just another way to say slope. Both tell you how much y changes when x changes by 1 unit.

Where is the y-intercept in this equation?

The y-intercept is 0 because there's no constant term. The equation $y = -6x$ means $y = -6x + 0$ , so the line passes through the origin (0,0).

How do I find rate of change if the equation looks different?

First, rearrange the equation to slope-intercept form $y = mx + b$ . Then the coefficient of x is your rate of change, no matter how the original equation was written!

Can rate of change be a fraction or decimal?

Absolutely! Rate of change can be any real number - positive, negative, whole numbers, fractions, or decimals. In this case, it's the whole number -6.

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