Find the Rate of Change: Analyzing y=1-4x Linear Function

Rate of Change with Negative Slope

Given the linear function:

y=14x y=1-4x

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 function's slope
00:04 Arrange the function to fit the formula
00:08 Use the formula to represent a linear function
00:14 We can see that the coefficient of X (M) is the function's slope
00:19 Find the function's slope using the coefficient of X
00:25 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Given the linear function:

y=14x y=1-4x

What is the rate of change of the function?

2

Step-by-step solution

The given linear function is y=14x y = 1 - 4x . This can be rearranged to fit the standard format of a linear equation, y=mx+b y = mx + b , as y=4x+1 y = -4x + 1 .

In this form, the slope m m is simply the coefficient of x x . Here, m=4 m = -4 .

The slope of the linear function represents the rate of change of the function with respect to x x , meaning for every unit increase in x x , the value of y y decreases by 4 units.

Therefore, the rate of change of the function is m=4 m = -4 , which is option 4 among the given choices.

Thus, the solution to the problem is m=4 m = -4 .

3

Final Answer

m=4 m=-4

Key Points to Remember

Essential concepts to master this topic
  • Definition: Rate of change equals the slope coefficient in linear functions
  • Technique: Rewrite y=14x y = 1 - 4x as y=4x+1 y = -4x + 1
  • Check: Slope is -4, so when x increases by 1, y decreases by 4 ✓

Common Mistakes

Avoid these frequent errors
  • Confusing y-intercept with slope
    Don't identify the constant term 1 as the rate of change = wrong answer! The y-intercept tells you where the line crosses the y-axis, not how steep it is. Always identify the coefficient of x as the slope and rate of change.

Practice Quiz

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

XY

FAQ

Everything you need to know about this question

What exactly is rate of change in a linear function?

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Rate of change is how much y changes when x increases by 1 unit. In y=mx+b y = mx + b , the slope m is always the rate of change.

Why is the rate of change negative here?

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A negative rate of change means the function is decreasing. As x gets larger, y gets smaller. The line slopes downward from left to right.

How do I rewrite y = 1 - 4x in standard form?

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Move the x term first: y=14x y = 1 - 4x becomes y=4x+1 y = -4x + 1 . Now you can easily see the slope is -4.

What does a rate of change of -4 mean practically?

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It means for every 1 unit increase in x, the y-value decreases by 4 units. The function drops 4 times faster than x increases.

Is rate of change the same as slope?

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Yes! For linear functions, rate of change and slope are exactly the same thing. Both describe how steep the line is and in which direction.

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