https://file.aiquickdraw.com/mathbot/file/1725838550778893rfutb.jpg

The graph shown is a transformation of the function \( f(x) = x^2 \). 

To determine the equation of the transformed function \( g(x) \), we need to identify the transformations applied to the basic parabola \( y = x^2 \).

### Analysis of the Graph:
1. **Vertex**: The vertex of the graph is located at \((-3, -4)\).
2. **Shape and Direction**: The graph is a parabola that opens upwards, which indicates no reflection over the x-axis.
3. **Horizontal Shift**: The graph is shifted 3 units to the left.
4. **Vertical Shift**: The graph is shifted 4 units down.

### Equation of the Transformation:
The standard form for a transformed quadratic function is:

\[
g(x) = a(x - h)^2 + k
\]

where:
- \( (h, k) \) is the vertex of the parabola,
- \( a \) determines the vertical stretch or compression and the direction of the parabola.

Since the vertex is at \((-3, -4)\), we have \( h = -3 \) and \( k = -4 \). There appears to be no vertical stretch or compression, so \( a = 1 \).

### Final Equation:
\[
g(x) = (x + 3)^2 - 4
\]

Would you like more details or have any questions?

### Related Questions:
1. How would the equation change if the parabola were reflected over the x-axis?
2. What would the equation look like if the graph were vertically stretched by a factor of 2?
3. How can you determine the transformations if given only the equation?
4. What is the effect of changing the value of \( a \) in the equation?
5. How do horizontal and vertical shifts affect the vertex form of a quadratic equation?

### Tip:
Always identify the vertex and the direction of opening first when analyzing the transformation of a parabola.

The graph above is a transformation of the function x^2. Write an equation for the function graphed above.

Learn how to write the equation of a quadratic function based on its graph, which is a transformation of x^2. This problem involves identifying shifts, determining the vertex, and applying the vertex form equation. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation