Math / Algebra

QuestionWhich of the following transformations have occurred to the absolute value parent function to create the graph of $k(x)=\frac{1}{2}|-(x-2)|-3$ ? Select all the apply. - Reflection over the x-axis. Reflection over the $y$ -axis. Vertical compression Vertical stretch $\qquad$ Horizontal compression Horizontal stretch Translation right Translation left Translation up Translation down

Studdy Solution

STEP 1

What is this asking? This problem is asking us to identify all the ways the basic absolute value graph, $|x|$ , has been changed to make the new graph $k(x) = \frac{1}{2}|-(x-2)| - 3$ . Watch out! Be careful with the signs inside and outside the absolute value!
They can be tricky!

STEP 2

1. Simplify the function
2. Analyze the transformations

STEP 3

We can rewrite the absolute value part of our function.
Remember that $|-\text{anything}| = |\text{anything}|$ .
So, we have $|-(x-2)| = |x-2|$ .
This makes our function $k(x) = \frac{1}{2}|x-2| - 3$ .

STEP 4

The $\frac{1}{2}$ in front of the absolute value tells us how much the graph is stretched or compressed vertically.
Since $\frac{1}{2}$ is between 0 and 1, it's a vertical compression by a factor of $\frac{1}{2}$ .
Imagine squishing the graph vertically towards the x-axis!

STEP 5

The $-2$ inside the absolute value with the $x$ tells us about a horizontal shift.
Since it's $(x-2)$ , we shift the graph 2 units to the right.
Remember, inside the absolute value, it's the opposite of what it looks like! $x - 2$ means move to the positive direction.

STEP 6

Finally, the $-3$ outside the absolute value tells us about a vertical shift.
Since it's $-3$ , we shift the graph 3 units down.

STEP 7

The transformations are: * Vertical compression * Translation right * Translation down

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Studdy Solution

STEP 1

What is this asking? This problem is asking us to identify all the ways the basic absolute value graph, $|x|$ , has been changed to make the new graph $k(x) = \frac{1}{2}|-(x-2)| - 3$ . Watch out! Be careful with the signs inside and outside the absolute value!
They can be tricky!

STEP 2

1. Simplify the function
2. Analyze the transformations

STEP 3

We can rewrite the absolute value part of our function.
Remember that $|-\text{anything}| = |\text{anything}|$ .
So, we have $|-(x-2)| = |x-2|$ .
This makes our function $k(x) = \frac{1}{2}|x-2| - 3$ .

STEP 4

The $\frac{1}{2}$ in front of the absolute value tells us how much the graph is stretched or compressed vertically.
Since $\frac{1}{2}$ is between 0 and 1, it's a vertical compression by a factor of $\frac{1}{2}$ .
Imagine squishing the graph vertically towards the x-axis!

STEP 5

The $-2$ inside the absolute value with the $x$ tells us about a horizontal shift.
Since it's $(x-2)$ , we shift the graph 2 units to the right.
Remember, inside the absolute value, it's the opposite of what it looks like! $x - 2$ means move to the positive direction.

STEP 6

Finally, the $-3$ outside the absolute value tells us about a vertical shift.
Since it's $-3$ , we shift the graph 3 units down.

STEP 7

The transformations are: * Vertical compression * Translation right * Translation down

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Studdy Solution

STEP 1

STEP 2

1. Simplify the function2. Analyze the transformations

STEP 3

STEP 4

The 12\frac{1}{2}21​ in front of the absolute value tells us how much the graph is stretched or compressed *vertically*.Since 12\frac{1}{2}21​ is between 0 and 1, it's a **vertical compression** by a factor of 12\frac{1}{2}21​.Imagine squishing the graph vertically towards the x-axis!

STEP 5

STEP 6

Finally, the −3-3−3 *outside* the absolute value tells us about a *vertical* shift.Since it's −3-3−3, we shift the graph **3 units down**.

STEP 7

The transformations are: * **Vertical compression** * **Translation right** * **Translation down**

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1. Simplify the function
2. Analyze the transformations

The $\frac{1}{2}$ in front of the absolute value tells us how much the graph is stretched or compressed vertically.
Since $\frac{1}{2}$ is between 0 and 1, it's a vertical compression by a factor of $\frac{1}{2}$ .
Imagine squishing the graph vertically towards the x-axis!

Finally, the $-3$ outside the absolute value tells us about a vertical shift.
Since it's $-3$ , we shift the graph 3 units down.

The transformations are: * Vertical compression * Translation right * Translation down