Question Describe the transformation of $f(x) = \frac{1}{2}x - 2$ to have slope 2 and $y$ -intercept -8.

Studdy Solution

STEP 1

Assumptions
1. The original function is $f(x) = \frac{1}{2} x - 2$ .
2. We want to transform the function to have a slope of 2.
3. We want the transformed function to have a $y$ -intercept of -8.
4. We need to identify which transformation among the given options (A, B, C, D) achieves the desired changes.

STEP 2

First, let's understand what each transformation does to the function $f(x)$ .
(A) $f(4x)$ - This transformation compresses the graph horizontally by a factor of 4. It does not change the slope or the $y$ -intercept.
(B) $4f(x)$ - This transformation multiplies the output of the function by 4, effectively quadrupling the slope, but it does not change the $y$ -intercept.
(C) $f(x) - 6$ - This transformation shifts the graph of the function downward by 6 units. It does not change the slope but changes the $y$ -intercept.
(D) $f(x-4)$ - This transformation shifts the graph of the function to the right by 4 units. It does not change the slope but changes the $y$ -intercept.

STEP 3

Now, we want to find a transformation that changes the slope to 2. The original slope is $\frac{1}{2}$ , so we need to multiply the slope by 4 to get a new slope of 2.

STEP 4

The only transformation that changes the slope is (B) $4f(x)$ . Let's apply this transformation to the original function:
$4f(x) = 4\left(\frac{1}{2} x - 2\right)$

STEP 5

Distribute the 4 across the terms inside the parentheses:
$4f(x) = 4 \cdot \frac{1}{2} x - 4 \cdot 2$

STEP 6

Simplify the expression:
$4f(x) = 2x - 8$

STEP 7

Now we have a function with a slope of 2. We also need to check if the $y$ -intercept is -8.

STEP 8

To find the $y$ -intercept, we set $x = 0$ in the transformed function:
$4f(0) = 2 \cdot 0 - 8$

STEP 9

Calculate the $y$ -intercept:
$4f(0) = -8$

STEP 10

The $y$ -intercept of the transformed function is indeed -8, which matches our requirement.

STEP 11

Since the transformation (B) $4f(x)$ changes the slope to 2 and the $y$ -intercept to -8, this is the correct transformation that achieves the desired changes to the function.
The correct transformation is (B) $4f(x)$ .

Was this helpful?

Question Describe the transformation of $f(x) = \frac{1}{2}x - 2$ to have slope 2 and $y$ -intercept -8.

Studdy Solution

STEP 1

STEP 2

STEP 3

Now, we want to find a transformation that changes the slope to 2. The original slope is $\frac{1}{2}$ , so we need to multiply the slope by 4 to get a new slope of 2.

STEP 4

The only transformation that changes the slope is (B) $4f(x)$ . Let's apply this transformation to the original function:
$4f(x) = 4\left(\frac{1}{2} x - 2\right)$

STEP 5

Distribute the 4 across the terms inside the parentheses:
$4f(x) = 4 \cdot \frac{1}{2} x - 4 \cdot 2$

STEP 6

Simplify the expression:
$4f(x) = 2x - 8$

STEP 7

Now we have a function with a slope of 2. We also need to check if the $y$ -intercept is -8.

STEP 8

To find the $y$ -intercept, we set $x = 0$ in the transformed function:
$4f(0) = 2 \cdot 0 - 8$

STEP 9

Calculate the $y$ -intercept:
$4f(0) = -8$

STEP 10

The $y$ -intercept of the transformed function is indeed -8, which matches our requirement.

STEP 11

Since the transformation (B) $4f(x)$ changes the slope to 2 and the $y$ -intercept to -8, this is the correct transformation that achieves the desired changes to the function.
The correct transformation is (B) $4f(x)$ .

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question Describe the transformation of f(x)=12x−2f(x) = \frac{1}{2}x - 2f(x)=21​x−2 to have slope 2 and yyy-intercept -8.

Studdy Solution

STEP 1

STEP 2

STEP 3

Now, we want to find a transformation that changes the slope to 2. The original slope is 12 \frac{1}{2} 21​, so we need to multiply the slope by 4 to get a new slope of 2.

STEP 4

The only transformation that changes the slope is (B) 4f(x) 4f(x) 4f(x). Let's apply this transformation to the original function:4f(x)=4(12x−2) 4f(x) = 4\left(\frac{1}{2} x - 2\right) 4f(x)=4(21​x−2)

STEP 5

Distribute the 4 across the terms inside the parentheses:4f(x)=4⋅12x−4⋅2 4f(x) = 4 \cdot \frac{1}{2} x - 4 \cdot 2 4f(x)=4⋅21​x−4⋅2

STEP 6

Simplify the expression:4f(x)=2x−8 4f(x) = 2x - 8 4f(x)=2x−8

STEP 7

Now we have a function with a slope of 2. We also need to check if the y y y-intercept is -8.

STEP 8

To find the y y y-intercept, we set x=0 x = 0 x=0 in the transformed function:4f(0)=2⋅0−8 4f(0) = 2 \cdot 0 - 8 4f(0)=2⋅0−8

STEP 9

Calculate the y y y-intercept:4f(0)=−8 4f(0) = -8 4f(0)=−8

STEP 10

The y y y-intercept of the transformed function is indeed -8, which matches our requirement.

STEP 11

Since the transformation (B) 4f(x) 4f(x) 4f(x) changes the slope to 2 and the y y y-intercept to -8, this is the correct transformation that achieves the desired changes to the function.The correct transformation is (B) 4f(x) 4f(x) 4f(x).

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Question Describe the transformation of $f(x) = \frac{1}{2}x - 2$ to have slope 2 and $y$ -intercept -8.

Now, we want to find a transformation that changes the slope to 2. The original slope is $\frac{1}{2}$ , so we need to multiply the slope by 4 to get a new slope of 2.

The only transformation that changes the slope is (B) $4f(x)$ . Let's apply this transformation to the original function:
$4f(x) = 4\left(\frac{1}{2} x - 2\right)$

Distribute the 4 across the terms inside the parentheses:
$4f(x) = 4 \cdot \frac{1}{2} x - 4 \cdot 2$

Simplify the expression:
$4f(x) = 2x - 8$

Now we have a function with a slope of 2. We also need to check if the $y$ -intercept is -8.

To find the $y$ -intercept, we set $x = 0$ in the transformed function:
$4f(0) = 2 \cdot 0 - 8$

Calculate the $y$ -intercept:
$4f(0) = -8$

The $y$ -intercept of the transformed function is indeed -8, which matches our requirement.

Since the transformation (B) $4f(x)$ changes the slope to 2 and the $y$ -intercept to -8, this is the correct transformation that achieves the desired changes to the function.
The correct transformation is (B) $4f(x)$ .