Calculate Rate of Change: Finding the Slope of y=16+16x

Q: What's the difference between slope and y-intercept?

The slope is how much y changes for each unit increase in x (the coefficient of x). The y-intercept is where the line crosses the y-axis (the constant term).

Q: Why is the rate of change the same as slope?

Rate of change and slope mean the exact same thing! Both describe how fast y increases or decreases as x increases by 1 unit.

Q: Can the rate of change be negative?

Yes! A negative slope means the line goes down from left to right. For example, in $ y = -3x + 10 $, the rate of change is -3.

Linear Functions with Coefficient Identification

Given the linear function:

$y=16+16x$

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 Let's arrange the equation

00:06 The rate of change is the slope of the function (M)

00:09 Let's use the straight line equation

00:16 Let's compare and find the slope of the function using the straight line equation

00:20 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:

$y=16+16x$

What is the rate of change of the function?

Step-by-step solution

To solve this problem, we'll follow these steps:

Identify the standard form of the linear function.
Determine the coefficient of $x$ , which is the slope $m$ .

Now, let's work through each step:

Step 1: The given linear function is $y = 16 + 16x$ , which is in the form $y = mx + b$ .

Step 2: In this function, the coefficient of $x$ is $16$ . This coefficient is the slope $m$ of the function.

Therefore, the rate of change of the function, or the slope, is $m = 16$ .

Final Answer

$m=16$

Key Points to Remember

Essential concepts to master this topic

Slope Formula: In y = mx + b, m is always the rate of change
Coefficient Method: For y = 16 + 16x, the coefficient 16 is the slope
Verification: Check that slope stays constant between any two points ✓

Common Mistakes

Avoid these frequent errors

Confusing the y-intercept with the slope
Don't identify 16 (the constant) as the slope = wrong rate of change! The constant term is the y-intercept, not the slope. Always look for the coefficient of x to find the slope.

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's the difference between slope and y-intercept?

The slope is how much y changes for each unit increase in x (the coefficient of x). The y-intercept is where the line crosses the y-axis (the constant term).

Why is the rate of change the same as slope?

Rate of change and slope mean the exact same thing! Both describe how fast y increases or decreases as x increases by 1 unit.

What if the equation is written as y = 16x + 16?

The slope is still 16! The order of terms doesn't matter - just identify the coefficient of x. Whether it's $y = 16 + 16x$ or $y = 16x + 16$ , the slope is always 16.

How do I find the slope if x has no visible coefficient?

If you see just x (like in y = x + 3), the coefficient is 1. If you see -x (like in y = -x + 5), the coefficient is -1.

Can the rate of change be negative?

Yes! A negative slope means the line goes down from left to right. For example, in $y = -3x + 10$ , the rate of change is -3.

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