Use transformations of f left parenthesis x right parenthesis equals x squared to graph the following function.
h left parenthesis x right parenthesis equals left parenthesis x minus 5 right parenthesis squared plus 2


To graph the function \( h(x) = (x - 5)^2 + 2 \) using transformations of the parent function \( f(x) = x^2 \), follow these steps:

### 1. **Start with the parent function:**
   The parent function is \( f(x) = x^2 \), which is a parabola that opens upwards with its vertex at the origin \((0, 0)\).

### 2. **Apply a horizontal shift:**
   The term \( (x - 5) \) indicates a horizontal shift. Specifically, the graph of \( f(x) = x^2 \) is shifted **5 units to the right**. 
   
   - New vertex: \((5, 0)\)

### 3. **Apply a vertical shift:**
   The \( +2 \) outside the squared term indicates a vertical shift. The entire graph is shifted **2 units up**.
   
   - Final vertex: \((5, 2)\)

### 4. **Graph the transformed function:**
   The graph of \( h(x) = (x - 5)^2 + 2 \) is a parabola that opens upwards with its vertex at \( (5, 2) \). The shape of the parabola remains the same as the original \( f(x) = x^2 \), but it is shifted right by 5 units and up by 2 units.

### Summary of transformations:
- **Right shift** by 5 units.
- **Upward shift** by 2 units.

Would you like details on how to plot this step by step, or do you have any questions?

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Here are 5 related questions for further practice:

1. How would the graph of \( h(x) = (x + 3)^2 - 4 \) be transformed?
2. What is the effect of the transformation \( h(x) = 2(x - 1)^2 + 3 \)?
3. How does adding a negative sign in front of the function, such as \( h(x) = -(x - 2)^2 + 1 \), change the graph?
4. How would you describe the transformation if the function was \( h(x) = \frac{1}{2}(x - 4)^2 - 3 \)?
5. What happens to the graph of \( f(x) = x^2 \) if you apply a vertical stretch?

**Tip:** Remember that horizontal shifts are determined by the expression inside the function with the variable \( x \), while vertical shifts are determined by constants added or subtracted outside the function.

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

Explore how to graph the function h(x) = (x - 5)^2 + 2 using transformations of the parent function f(x) = x^2. This problem highlights key concepts in graphing quadratic functions, including horizontal and vertical shifts. Perfect for students in Grades 9-11.

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