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

What is the solution to the following inequality?

$$ 10x-4\leq-3x-8 $$

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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