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

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

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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