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Graph the equation shown below by transforming the given graph of the parent function. $y=\frac{1}{3}|x|$ Start Over

Studdy Solution

STEP 1

What is this asking? We need to graph $y = \frac{1}{3}|x|$ by changing the basic shape of $y = |x|$ . Watch out! Don't mix up inside and outside changes to the function!
Inside changes affect the x-values, and outside changes affect the y-values.

STEP 2

1. Understand the parent function
2. Apply the transformation

STEP 3

Alright, let's kick things off with the absolute value function, $y = |x|$ !
This is our parent function, the foundation of our graph!
It's super important to understand what it looks like.

STEP 4

Remember, the absolute value of a number is its distance from zero.
So, $|x|$ is just asking "how far is $x$ from zero?".

STEP 5

Let's plot a few points to see the shape!
If $x = -2$ , then $y = |-2| = 2$ .
If $x = -1$ , then $y = |-1| = 1$ .
If $x = 0$ , then $y = |0| = 0$ .
If $x = 1$ , then $y = |1| = 1$ .
If $x = 2$ , then $y = |2| = 2$ .

STEP 6

When we connect these points, we get a V-shape centered at the origin!
It's like a boomerang that always comes back positive!

STEP 7

Now, let's look at our equation: $y = \frac{1}{3}|x|$ .
What's different from the parent function?
We're multiplying the absolute value of $x$ by $\frac{1}{3}$ .

STEP 8

Since the $\frac{1}{3}$ is outside the absolute value, it affects the y-values.
This is a vertical compression!
It's like we're squishing the graph down towards the x-axis.

STEP 9

Think of it this way: for every x-value, the corresponding y-value is now one-third of what it used to be in the parent function.
So, if the parent function had the point $(2, 2)$ , our new function will have the point $(2, \frac{1}{3} \cdot 2)$ , which simplifies to $(2, \frac{2}{3})$ .

STEP 10

Let's plot some points to see this in action!
If $x = -3$ , then $y = \frac{1}{3}|-3| = \frac{1}{3} \cdot 3 = 1$ .
If $x = 0$ , then $y = \frac{1}{3}|0| = \frac{1}{3} \cdot 0 = 0$ .
If $x = 3$ , then $y = \frac{1}{3}|3| = \frac{1}{3} \cdot 3 = 1$ .

STEP 11

Connect these points, and you'll see our vertically compressed V-shape!
It's still centered at the origin, but it's wider than the parent function because the y-values are smaller for the same x-values.

STEP 12

The graph of $y = \frac{1}{3}|x|$ is a V-shape centered at the origin, vertically compressed by a factor of $\frac{1}{3}$ compared to the parent function $y = |x|$ .

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