Question $y = 3\left|2 - \frac{x}{3}\right|$

Studdy Solution

STEP 1

What is this asking? This is asking us to understand what this absolute value equation looks like! Watch out! Absolute value can be tricky, so let's make sure we don't forget how it behaves!

STEP 2

1. Understand the absolute value
2. Analyze the equation

STEP 3

Alright, so we've got this equation $y = 3\left|2 - \frac{x}{3}\right|$ .
Remember, absolute value means whatever is inside those vertical bars becomes positive!
So, if $2 - \frac{x}{3}$ is positive, it stays the same.
If it's negative, we multiply it by $-1$ to make it positive!

STEP 4

The "pointy part" of the absolute value graph, the vertex, happens when the stuff inside the absolute value equals zero.
So, let's set $2 - \frac{x}{3} = 0$ .
To solve for $x$ , let's add $\frac{x}{3}$ to both sides to get $2 = \frac{x}{3}$ .
Then, we multiply both sides by $3$ to get $x = 6$ .
When $x = 6$ , $y = 3|2 - \frac{6}{3}| = 3|2 - 2| = 3 \cdot 0 = 0$ .
So, our vertex is at the point $(6, 0)$ .

STEP 5

When $2 - \frac{x}{3} > 0$ , the absolute value doesn't do anything!
So, we have $y = 3(2 - \frac{x}{3})$ .
Distributing the $3$ , we get $y = 6 - x$ .
This is a line with a slope of $-1$ and a y-intercept of $6$ .
This happens when $x < 6$ .

STEP 6

When $2 - \frac{x}{3} < 0$ , the absolute value flips the sign!
So, we have $y = 3(-1)(2 - \frac{x}{3}) = 3(\frac{x}{3} - 2) = x - 6$ .
This is a line with a slope of $1$ and a y-intercept of $-6$ .
This happens when $x > 6$ .

STEP 7

Imagine the graph.
It's a "V" shape with the point at $(6, 0)$ .
To the left of the vertex, it's the line $y = 6 - x$ .
To the right of the vertex, it's the line $y = x - 6$ .

STEP 8

The graph of the equation $y = 3\left|2 - \frac{x}{3}\right|$ is a "V" shape with its vertex at $(6, 0)$ .
For $x < 6$ , the graph follows the line $y = 6 - x$ .
For $x > 6$ , it follows the line $y = x - 6$ .

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