Linear Function: Plotting Points (36,-60) and (6,30) on a Graph

Q: Why does the slope being negative mean the function is decreasing?

A negative slope means that as x-values increase, y-values decrease. Think of it like going downhill - as you move right (positive x-direction), you go down (negative y-direction).

Q: What would a constant function look like?

A constant function has slope = 0, meaning the y-values never change. Both points would have the same y-coordinate, like (3, 5) and (8, 5).

Q: How can I remember the slope formula?

Think "rise over run" - the rise is the change in y-values (vertical), and the run is the change in x-values (horizontal). So: $ \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} $

Slope Calculation with Two Points

The graph of the linear function passes through the points $B(36,-60),A(6,30)$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

The graph of the linear function passes through the points $B(36,-60),A(6,30)$

Step-by-step solution

To solve this problem, we will compute the slope of the line that passes through the points $A(6, 30)$ and $B(36, -60)$ .

Step 1: Apply the slope formula
The slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is computed as follows:

$m = \frac{y_2 - y_1}{x_2 - x_1}$

Substituting the values for points $A(6, 30)$ and $B(36, -60)$ :
$m = \frac{-60 - 30}{36 - 6} = \frac{-90}{30} = -3$

Step 2: Analyze the slope
Since the slope $m = -3$ is negative, it indicates that the linear function is decreasing.

Therefore, the solution to the problem is a decreasing function.

Final Answer

Decreasing function

Key Points to Remember

Essential concepts to master this topic

Slope Formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ for points $(x_1, y_1)$ and $(x_2, y_2)$
Technique: Substitute A(6,30) and B(36,-60): $m = \frac{-60-30}{36-6} = -3$
Check: Negative slope means y decreases as x increases ✓

Common Mistakes

Avoid these frequent errors

Confusing the order of coordinates when subtracting
Don't mix up which point is (x₁, y₁) and which is (x₂, y₂) = wrong slope sign! This changes whether your function increases or decreases. Always keep the same point order: if B is second, use B's coordinates as (x₂, y₂) throughout.

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

Why does the slope being negative mean the function is decreasing?

A negative slope means that as x-values increase, y-values decrease. Think of it like going downhill - as you move right (positive x-direction), you go down (negative y-direction).

Does it matter which point I call A and which I call B?

No! The slope will be the same regardless of order. Just make sure you're consistent - if you use A's coordinates first in the numerator, use A's coordinates first in the denominator too.

What would a constant function look like?

A constant function has slope = 0, meaning the y-values never change. Both points would have the same y-coordinate, like (3, 5) and (8, 5).

How can I remember the slope formula?

Think "rise over run" - the rise is the change in y-values (vertical), and the run is the change in x-values (horizontal). So: $\frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x}$

What if I get a positive slope instead?

Double-check your subtraction! Make sure you're subtracting in the correct order. With points A(6,30) and B(36,-60), you should get $\frac{-60-30}{36-6} = \frac{-90}{30} = -3$ .

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