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.