Find the Rate of Change: Analyzing y=14x+13 Linear Function

Q: What exactly does 'rate of change' mean in a linear function?

Rate of change tells you how much y increases for every 1-unit increase in x. In $ y = 14x + 13 $, when x goes up by 1, y goes up by 14!

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

Because slope measures steepness, which is exactly the rate of change! A steeper line means y changes faster as x increases, so slope = rate of change.

Q: How can I remember which number is the rate of change?

Easy trick: In $ y = mx + b $, the rate of change is always the number multiplied by x. The lone number (b) is just where the line starts on the y-axis.

Q: What if the coefficient of x was negative, like -14?

Then the rate of change would be -14! This means y decreases by 14 units for every 1-unit increase in x. Negative rates show decreasing functions.

Q: Can I find the rate of change if the equation isn't in y = mx + b form?

Yes! Just rearrange the equation to solve for y first. Once you have $ y = mx + b $ form, the coefficient of x is your rate of change.

Linear Functions with Slope-Intercept Form

Given the linear function:

$y=14x+13$

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

00:08 Let's find the rate of change in this function.

00:11 The rate of change is also known as the slope of the function.

00:16 To do this, we'll use the equation of a straight line.

00:20 We'll compare it and find the slope using this equation.

00:24 And there you have it, that's how we solve this question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the linear function:

$y=14x+13$

What is the rate of change of the function?

Step-by-step solution

To solve this problem, we need to find the rate of change, which is represented by the slope of the linear function.

The function provided is in the form $y = mx + b$ , where $m$ is the slope. This is known as the slope-intercept form of a linear equation.

Given the equation $y = 14x + 13$ , we can directly identify that the coefficient of $x$ , which is 14, represents the slope $m$ , or the rate of change of the function.

Therefore, the rate of change, or the slope, of this function is $m = 14$ .

Final Answer

$m=14$

Key Points to Remember

Essential concepts to master this topic

Slope-Intercept Form: In $y = mx + b$ , m is the rate of change
Identification: The coefficient of x equals the slope: 14x means m = 14
Verification: Rate of change = 14 means y increases 14 units per x unit ✓

Common Mistakes

Avoid these frequent errors

Confusing the y-intercept with the rate of change
Don't identify the constant term (13) as the rate of change = completely wrong slope! The y-intercept shows 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

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

FAQ

Everything you need to know about this question

What exactly does 'rate of change' mean in a linear function?

Rate of change tells you how much y increases for every 1-unit increase in x. In $y = 14x + 13$ , when x goes up by 1, y goes up by 14!

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

Because slope measures steepness, which is exactly the rate of change! A steeper line means y changes faster as x increases, so slope = rate of change.

How can I remember which number is the rate of change?

Easy trick: In $y = mx + b$ , the rate of change is always the number multiplied by x. The lone number (b) is just where the line starts on the y-axis.

What if the coefficient of x was negative, like -14?

Then the rate of change would be -14! This means y decreases by 14 units for every 1-unit increase in x. Negative rates show decreasing functions.

Can I find the rate of change if the equation isn't in y = mx + b form?

Yes! Just rearrange the equation to solve for y first. Once you have $y = mx + b$ form, the coefficient of x is your rate of change.

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