Use transformations of f left parenthesis x right parenthesis equals x squared to graph the following function.
g left parenthesis x right parenthesis equals negative 3 left parenthesis x plus 3 right parenthesis squared plus 1
Question content area bottom left
Part 1
What transformations are needed to graph the function g left parenthesis x right parenthesis equals negative 3 left parenthesis x plus 3 right parenthesis squared plus 1​? Choose the correct answer below.
A.
The graph of f left parenthesis x right parenthesis equals x squared should be horizontally shifted to the right by 3 units​, stretched vertically by a factor of 3​, reflected about the​ y-axis, and shifted vertically up by 1 unit.
B.
The graph of f left parenthesis x right parenthesis equals x squared should be horizontally shifted to the right by 3 units​, stretched horizontally by a factor of 3​, reflected about the​ y-axis, and shifted vertically up by 1 unit.
C.
The graph of f left parenthesis x right parenthesis equals x squared should be horizontally shifted to the right by 3 units​, stretched horizontally by a factor of 3​, reflected about the​ x-axis, and shifted vertically up by 1 unit.
D.
The graph of f left parenthesis x right parenthesis equals x squared should be horizontally shifted to the left by 3 units​, stretched vertically by a factor of 3​, reflected about the​ x-axis, and shifted vertically up by 1 unit.

To solve this problem, we need to understand the transformations applied to the function \( f(x) = x^2 \) in order to obtain the function \( g(x) = -3(x + 3)^2 + 1 \).

### Steps to determine the transformations:
1. **Start with the basic function**: \( f(x) = x^2 \).
2. **Horizontal Shift**: The expression \( (x + 3) \) inside the square indicates a horizontal shift to the left by 3 units.
3. **Vertical Stretch/Compression**: The coefficient \( -3 \) outside the squared term indicates two transformations:
   - The negative sign reflects the graph across the \( x \)-axis.
   - The factor of 3 indicates a vertical stretch by a factor of 3.
4. **Vertical Shift**: The \( +1 \) outside the squared term indicates a vertical shift upwards by 1 unit.

### Summary of transformations:
- **Horizontal shift** to the left by 3 units.
- **Vertical stretch** by a factor of 3.
- **Reflection** across the \( x \)-axis.
- **Vertical shift** up by 1 unit.

### Correct Answer:
The correct answer is **D**:
- "The graph of \( f(x) = x^2 \) should be horizontally shifted to the left by 3 units, stretched vertically by a factor of 3, reflected about the \( x \)-axis, and shifted vertically up by 1 unit."

Would you like more details on any specific transformation, or do you have any questions?

Here are 5 related questions to expand on this:
1. How does a horizontal shift affect the graph of a function?
2. What is the impact of a vertical stretch on the shape of the graph?
3. How does reflecting a function across the \( x \)-axis change its graph?
4. What are the effects of combining multiple transformations on a function?
5. How would the transformations change if the function was \( g(x) = -\frac{1}{2}(x - 3)^2 - 2 \)?

**Tip:** When dealing with transformations, always apply horizontal shifts and reflections before vertical stretches/compressions and vertical shifts for accurate graphing.

Learn how to graph the function g(x) = -3(x + 3)^2 + 1 by understanding the transformations applied to the basic quadratic function. This problem covers horizontal shifts, vertical stretches, reflections across the x-axis, and vertical shifts. Suitable for high school students studying transformations of functions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation