Analyzing the Linear Function y = 3 - x: Increasing or Decreasing?

Linear Functions with Negative Slopes

Given the following function:

y=3x y=3-x

Is the function increasing or decreasing?

–4–4–4–2–2–2222444–2–2–2222000

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:07 Is the function increasing or decreasing? Let's find out!
00:11 Here's the function equation we have based on the given data.
00:15 We'll be using the linear equation for this.
00:18 Let's arrange our equation to fit the linear equation format.
00:23 Remember, minus is like multiplying by negative 1.
00:27 The slope of the function is negative, just like the data shows.
00:33 If the slope is negative, the function is decreasing.
00:37 And that's the solution to our question!

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Given the following function:

y=3x y=3-x

Is the function increasing or decreasing?

–4–4–4–2–2–2222444–2–2–2222000

2

Step-by-step solution

To determine whether the function y=3x y = 3 - x is increasing or decreasing, we need to examine its slope.

The given function is in the standard linear form y=mx+b y = mx + b , where m m is the slope. For the function y=3x y = 3 - x , the slope m m is 1-1.

The behavior of a linear function is determined by its slope:

  • If the slope m>0 m > 0 , the function is increasing.
  • If the slope m<0 m < 0 , the function is decreasing.

In this case, since the slope m=1 m = -1 , which is less than zero, the function is decreasing.

Thus, the function y=3x y = 3 - x is decreasing.

The correct choice is choice 2: Decreasing.

3

Final Answer

Decreasing

Key Points to Remember

Essential concepts to master this topic
  • Slope Rule: Negative slope means function decreases as x increases
  • Technique: Rewrite y=3x y = 3 - x as y=x+3 y = -x + 3 to identify slope
  • Check: Pick two points: when x = 0, y = 3; when x = 3, y = 0 (decreasing) ✓

Common Mistakes

Avoid these frequent errors
  • Confusing the y-intercept with the slope
    Don't look at the constant term 3 to determine if the function increases or decreases = wrong conclusion! The constant is just where the line crosses the y-axis. Always identify the coefficient of x (which is -1) to find the slope.

Practice Quiz

Test your knowledge with interactive questions

Is the function in the graph decreasing? yx

FAQ

Everything you need to know about this question

How do I quickly identify the slope in y = 3 - x?

+

Rewrite the equation in slope-intercept form y=mx+b y = mx + b . So y=3x y = 3 - x becomes y=1x+3 y = -1x + 3 , making the slope m = -1.

What does a slope of -1 mean visually?

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A slope of -1 means for every 1 unit you move right on the x-axis, the line goes down 1 unit on the y-axis. It's a 45-degree downward slant!

Why is this function decreasing and not increasing?

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Because the slope is negative (-1). When slope is negative, as x gets larger, y gets smaller. That's the definition of a decreasing function!

Can I tell if it's increasing or decreasing just by looking at the graph?

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Yes! Look at the line from left to right. If it goes up, it's increasing. If it goes down, it's decreasing. This line clearly slopes downward.

What if the equation was y = x + 3 instead?

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Then the slope would be positive (+1), making it an increasing function. The sign of the slope is what determines increasing vs decreasing behavior.

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