Question
For a parent function \( f(x) \). Write the transformed function \( g(x) \) for each description in the form \[ g(x)=A \cdot f(B(x-C)+D \] Vertical translation down 2 units Horizontal stretch by a factor of 3
Ask by Mccarthy Matthews. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The transformed function \( g(x) \) is:
\[
g(x) = f\left(\frac{1}{3}x\right) - 2
\]
Solution
To apply the given transformations to the parent function \( f(x) \) and express the transformed function \( g(x) \) in the form
\[
g(x) = A \cdot f\left(B(x - C)\right) + D,
\]
let's break down each transformation:
1. **Vertical Translation Down 2 Units:**
- This moves the graph of the function 2 units downward.
- **Effect on \( D \):** \( D = -2 \)
2. **Horizontal Stretch by a Factor of 3:**
- A horizontal stretch by a factor of 3 means that the graph is stretched away from the y-axis by 3 units. To achieve this, you scale the input \( x \) by a factor of \( \frac{1}{3} \).
- **Effect on \( B \):** \( B = \frac{1}{3} \)
3. **No Vertical or Horizontal Shifts:**
- Since there are no additional vertical or horizontal shifts beyond the translations mentioned, \( A = 1 \) and \( C = 0 \).
Putting it all together, the transformed function \( g(x) \) is:
\[
g(x) = f\left(\frac{1}{3}x\right) - 2
\]
**Final Expression:**
\[
g(x) = f\left(\frac{1}{3}x\right) - 2
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To create the transformed function \( g(x) \), start with the parent function \( f(x) \). For a vertical translation down 2 units, you subtract 2 from the entire function, giving you \( D = -2 \). For a horizontal stretch by a factor of 3, you modify the input \( x \) by dividing by 3, which makes \( B = \frac{1}{3} \). Since there’s no horizontal shift, \( C = 0 \). So, your function looks like this: \[ g(x) = A \cdot f\left(\frac{1}{3}(x-0)\right) - 2 \] Where \( A \) can remain 1 if no vertical scaling is applied, thus we can simplify it to: \[ g(x) = f\left(\frac{1}{3}x\right) - 2 \]
