https://file.aiquickdraw.com/mathbot/file/17296451536985faxdjof.jpg

The problem asks you to write an equation for the function in blue, which is a transformation of the basic function \( y = x^2 \), represented by the green graph.

Let's analyze the transformation:

1. **Vertical shift**: 
   - The blue parabola's vertex is higher than that of the green parabola. The green parabola has a vertex at \( (0,0) \), and the blue one appears to have a vertex at \( (0, 5) \). So, the blue graph is shifted **up by 5 units**.

2. **Horizontal shift**: 
   - Both graphs share the same x-coordinate for their vertex, at \( x = 0 \). There is **no horizontal shift**.

3. **Vertical stretch**: 
   - The blue graph is **narrower** than the green graph, indicating a vertical stretch. To determine the factor, we can compare the y-values at a specific x-coordinate. 
   - For example, at \( x = 1 \), the green parabola has \( y = 1^2 = 1 \), while the blue parabola has \( y = 10 \). Therefore, the blue parabola has been stretched vertically by a factor of 10.

### Equation:

The general form of the transformed function is:

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

Where \( (h, k) \) is the vertex, and \( a \) is the vertical stretch factor.

- \( h = 0 \) (no horizontal shift)
- \( k = 5 \) (shift up by 5)
- \( a = 10 \) (vertical stretch by a factor of 10)

Thus, the equation of the blue graph is:

\[
y = 10x^2 + 5
\]

Let me know if you'd like further details or have questions!

Here are 5 related questions:
1. What is the effect of a negative value for \( a \) in \( y = a(x - h)^2 + k \)?
2. How does a horizontal shift affect the graph of a quadratic function?
3. What happens to the graph if \( k \) is negative instead of positive?
4. Can you describe the effect of fractional values of \( a \) on the graph of \( y = x^2 \)?
5. How would you reflect the graph across the x-axis?

**Tip**: For a transformation, always examine the vertex and the points on the graph to identify shifts and stretches.

Write an equation that represents the function graphed in blue by using transformations from y = x^2, which is graphed in green.

Explore how to determine the equation of a transformed quadratic function. This problem demonstrates vertical shifts, stretches, and the transformation equation y = a(x - h)^2 + k, suitable for Grades 9-12 students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation