https://file.aiquickdraw.com/mathbot/file/1728706582920kvtin2cw.jpg

Let's analyze the given graph, which is a transformation of the parent function \( x^2 \). The parent function of a parabola, \( f(x) = x^2 \), has a vertex at \( (0, 0) \) and opens upwards. From the graph you provided, we can identify the following transformations:

1. **Vertical shift**: The vertex of the parabola has moved to \( (-2, 3) \) from the origin. This indicates a **horizontal shift** of 2 units to the left and a **vertical shift** of 3 units up.
   
2. **Reflection**: The parabola opens downward, suggesting a reflection over the x-axis.

3. **Vertical stretch or compression**: There seems to be no vertical stretch or compression based on the scale of the graph, so the vertical scaling factor remains unchanged.

Given these transformations, the general form for a transformed quadratic function is:
\[
g(x) = a(x - h)^2 + k
\]
where \( (h, k) \) is the vertex, and \( a \) controls the vertical stretch and the direction of the parabola.

- The vertex is \( (-2, 3) \), so \( h = -2 \) and \( k = 3 \).
- Since the parabola is reflected over the x-axis, \( a = -1 \).

Thus, the equation for the function is:
\[
g(x) = -(x + 2)^2 + 3
\]

Would you like further details on this or have any questions? Here are some related questions for deeper understanding:

1. How does changing the value of \( a \) affect the shape of a parabola?
2. How would the equation change if the parabola were stretched vertically?
3. What happens to the vertex of the parabola if you change the value of \( h \)?
4. Can you describe what happens if only \( k \) changes in the equation of a parabola?
5. How does the horizontal shift impact the graph of a quadratic function?

**Tip:** A reflection over the x-axis occurs when the leading coefficient \( a \) in the quadratic equation is negative.

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

Understand how to find the equation of a transformed quadratic function using vertex form. The parabola undergoes a reflection over the x-axis and shifts horizontally and vertically. Suitable for 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