Find the Rate of Change in y = 1/4x + 8: Linear Equation Analysis

Q: Why is the rate of change 1/4 and not 8?

In the equation $ y = \frac{1}{4}x + 8 $, the coefficient of x (which is $ \frac{1}{4} $) tells us the rate of change. The 8 is the y-intercept, showing where the line crosses the y-axis.

Q: What does a rate of change of 1/4 actually mean?

A rate of change of $ \frac{1}{4} $ means for every 4 units you move right (horizontally), the line goes up 1 unit (vertically). It's a gentle upward slope!

Q: How is rate of change different from slope?

They're the same thing! Rate of change and slope both describe how steep a line is. In linear equations, we use these terms interchangeably.

Q: What if the rate of change was negative?

A negative rate of change means the line slopes downward from left to right. For example, if m = -2, the line goes down 2 units for every 1 unit right.

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

Yes! First solve for y to get it into slope-intercept form. Then the coefficient of x will be your rate of change.

Slope-Intercept Form with Rate of Change

For the following straight line equation, state what is the rate of change?

$y=\frac{1}{4}x+8$

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Step-by-step video solution

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00:00 What is the rate of change of the function?

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

00:06 And this is the solution to the question

Step-by-step written solution

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Understand the problem

For the following straight line equation, state what is the rate of change?

$y=\frac{1}{4}x+8$

Step-by-step solution

To determine the rate of change for the given line equation $y = \frac{1}{4}x + 8$ :

The equation is in the slope-intercept form, $y = mx + b$ , where $m$ represents the slope or the rate of change.
By comparing the given equation $y = \frac{1}{4}x + 8$ with the general form $y = mx + b$ , we see that the slope $m$ is $\frac{1}{4}$ .
The rate of change for this line, thus, is $\frac{1}{4}$ .

Therefore, the rate of change for the line is $\frac{1}{4}$ .

Final Answer

$\frac{1}{4}$

Key Points to Remember

Essential concepts to master this topic

Slope-Intercept Form: In y = mx + b, m is the rate of change
Identification: For $y = \frac{1}{4}x + 8$ , slope m = $\frac{1}{4}$
Check: Rate of change means rise over run = $\frac{1}{4}$ units up per 1 unit right ✓

Common Mistakes

Avoid these frequent errors

Confusing the y-intercept with the rate of change
Don't think the rate of change is 8 because it's the larger number = wrong identification! The y-intercept (8) shows where the line crosses the y-axis, not how steep it is. Always remember the coefficient of x is the rate of change.

Practice Quiz

Test your knowledge with interactive questions

Look at the graph below and determine whether the function's rate of change is constant or not:

FAQ

Everything you need to know about this question

Why is the rate of change 1/4 and not 8?

In the equation $y = \frac{1}{4}x + 8$ , the coefficient of x (which is $\frac{1}{4}$ ) tells us the rate of change. The 8 is the y-intercept, showing where the line crosses the y-axis.

What does a rate of change of 1/4 actually mean?

A rate of change of $\frac{1}{4}$ means for every 4 units you move right (horizontally), the line goes up 1 unit (vertically). It's a gentle upward slope!

How is rate of change different from slope?

They're the same thing! Rate of change and slope both describe how steep a line is. In linear equations, we use these terms interchangeably.

What if the rate of change was negative?

A negative rate of change means the line slopes downward from left to right. For example, if m = -2, the line goes down 2 units for every 1 unit right.

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

Yes! First solve for y to get it into slope-intercept form. Then the coefficient of x will be your rate of change.

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