https://file.aiquickdraw.com/mathbot/file/1728918746644xfulz4s2.jpg

The question asks for the transformations needed to graph the function \( h(x) = (x + 2)^2 + 5 \) using the basic quadratic function \( f(x) = x^2 \) as a starting point. 

We can break this down into two transformations:

1. **Horizontal Shift**: The term \( (x + 2)^2 \) represents a shift **2 units to the left**. This is because adding \( 2 \) inside the parentheses moves the graph in the negative \( x \)-direction.
  
2. **Vertical Shift**: The \( +5 \) outside the squared term represents a shift **5 units up**. This shifts the graph upward by 5 units on the \( y \)-axis.

So, the required transformations are:
- A **horizontal shift 2 units to the left**.
- A **vertical shift 5 units up**.

Would you like more details or have any questions on this?

Here are 5 related questions for you to consider:
1. What happens to the graph if the constant inside the parentheses is negative, like \( h(x) = (x - 3)^2 + 5 \)?
2. How would the graph change if the \( +5 \) were replaced with a negative value, like \( h(x) = (x + 2)^2 - 3 \)?
3. How do horizontal and vertical stretches or compressions affect the transformation of \( f(x) = x^2 \)?
4. Can you describe how to graph \( h(x) = -(x+2)^2 + 5 \) based on the transformations?
5. How would you graph \( h(x) = 2(x+2)^2 + 5 \), where the quadratic term is multiplied by a coefficient?

**Tip**: When applying transformations, always follow the order: horizontal transformations, then vertical transformations.

Use transformations of f(x) = x^2 to graph the following function. h(x) = (x + 2)^2 + 5

Learn how to apply horizontal and vertical transformations to graph the quadratic function h(x) = (x + 2)^2 + 5. This problem involves shifting the basic quadratic function f(x) = x^2 two units left and five units up. 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