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Math  /  Geometry

QuestionWhat is the rate of change of yy with respect to xx shown in the graph?
The rate of change is □\square

Studdy Solution

STEP 1

What is this asking? We need to find the slope of the line on this graph, which tells us how much yy changes for every change in xx. Watch out! Make sure you put the change in yy in the numerator and the change in xx in the denominator, and double check the signs!

STEP 2

1. Find two points
2. Calculate the rise
3. Calculate the run
4. Calculate the slope

STEP 3

Alright, let's **pick two points** right off the graph!
We can use (−6,6)(-6, 6) and (6,−6)(6, -6).
Easy peasy!

STEP 4

The **rise** is the change in the yy values.
We calculate it by subtracting the yy-value of the **first point** from the yy-value of the **second point**:
(−6)−(6)=−12 (-6) - (6) = -12 So our **rise** is −12-12.
That means we're going *down* 12 units!

STEP 5

The **run** is the change in the xx values.
We find it by subtracting the xx-value of the **first point** from the xx-value of the **second point**:
(6)−(−6)=6+6=12 (6) - (-6) = 6 + 6 = 12 Our **run** is 1212.
We're moving 1212 units to the *right*!

STEP 6

Now, the moment of truth!
The **slope**, or rate of change, is simply the **rise** divided by the **run**.
slope=riserun=−1212 \text{slope} = \frac{\text{rise}}{\text{run}} = \frac{-12}{12}

STEP 7

We can simplify this fraction by dividing both the numerator and the denominator by 1212:
−1212=−12⋅112⋅1=−1⋅121⋅12=−11⋅1212=−11⋅1=−1 \frac{-12}{12} = \frac{-12 \cdot 1}{12 \cdot 1} = \frac{-1 \cdot 12}{1 \cdot 12} = \frac{-1}{1} \cdot \frac{12}{12} = \frac{-1}{1} \cdot 1 = -1 So, our **slope** is −1-1!

STEP 8

The rate of change of yy with respect to xx is −1-1.

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