Let f (x) = x2. Write the equation for the final transformed graph if the following transformations are
applied to f : stretch vertically by a factor of 2; shift downward by 5 units; shift right by 4 units.
(Start with how we would write the new function, for example 5f (x − 1) + 9, and then expand the
answer.)

We are given the function \( f(x) = x^2 \) and asked to apply three transformations:

1. **Stretch vertically by a factor of 2**: This transformation multiplies the function by 2, changing \( f(x) \) to \( 2f(x) \), or \( 2x^2 \).

2. **Shift downward by 5 units**: This shifts the entire graph down by 5, so we subtract 5 from the function. The new function becomes \( 2x^2 - 5 \).

3. **Shift right by 4 units**: This shift requires us to adjust the input of the function by subtracting 4 from \( x \). The new function becomes \( 2(x - 4)^2 - 5 \).

### Final Transformed Function:
The equation for the final transformed graph is:
\[
g(x) = 2(x - 4)^2 - 5
\]

### Expanded Form:
Now, let's expand this equation:
\[
g(x) = 2(x - 4)^2 - 5
\]
First, expand \( (x - 4)^2 \):
\[
(x - 4)^2 = x^2 - 8x + 16
\]
Now substitute it back into the equation:
\[
g(x) = 2(x^2 - 8x + 16) - 5
\]
Distribute the 2:
\[
g(x) = 2x^2 - 16x + 32 - 5
\]
Simplify:
\[
g(x) = 2x^2 - 16x + 27
\]

So the final expanded form of the transformed function is:
\[
g(x) = 2x^2 - 16x + 27
\]

Would you like to go over any step in detail or have further questions?

Here are 5 related questions for you:
1. How do vertical stretches affect the graph of a quadratic function?
2. What happens to the vertex of a quadratic function when it is shifted horizontally?
3. How would a horizontal stretch by a factor of 3 affect the function \( f(x) = x^2 \)?
4. Can you describe the effect of a vertical reflection on \( f(x) = x^2 \)?
5. How does a vertical shift compare to a horizontal shift in terms of function transformations?

**Tip**: When performing multiple transformations, always handle shifts inside the parentheses first, followed by stretches or compressions, and finally vertical shifts.

Let f(x) = x^2. Write the equation for the final transformed graph if the following transformations are applied to f: stretch vertically by a factor of 2; shift downward by 5 units; shift right by 4 units.

Learn how to transform the graph of f(x) = x^2 by applying a vertical stretch by a factor of 2, shifting right by 4 units, and shifting downward by 5 units. The step-by-step solution covers expanding and simplifying quadratic functions. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation