https://file.aiquickdraw.com/mathbot/file/1728941880894002izbvh.jpg

The problem provided in the image asks you to perform transformations on the function \( f(x) = x^2 \). Specifically:

1. Shift the function upward by 34 units.
2. Shift the function 59 units to the right.

To perform these transformations, we modify the function \( f(x) \) as follows:

### 1. Vertical Shift (upward by 34 units):
To shift a function upward, we add the desired number of units to the output. So, for the vertical shift of 34 units:

\[
f(x) = x^2 \quad \text{becomes} \quad x^2 + 34
\]

### 2. Horizontal Shift (to the right by 59 units):
To shift a function to the right, we replace \( x \) with \( (x - h) \), where \( h \) is the number of units to shift. Since we are shifting 59 units to the right:

\[
f(x) = x^2 \quad \text{becomes} \quad (x - 59)^2
\]

### Combining Both Transformations:
We now combine both the horizontal and vertical shifts:

\[
g(x) = (x - 59)^2 + 34
\]

Thus, the new function is:

\[
g(x) = (x - 59)^2 + 34
\]

Would you like a detailed breakdown of how these transformations work or have any further questions?

Here are 5 related questions you could explore:
1. How do horizontal shifts differ from vertical shifts in function transformations?
2. What is the effect of shifting a function downward instead of upward?
3. How would you shift the function \( f(x) = x^2 \) left by 10 units and downward by 5 units?
4. What happens to the graph of \( f(x) = x^2 \) when reflected across the y-axis?
5. How does the vertex of a parabola change under horizontal and vertical shifts?

**Tip:** For horizontal shifts, remember to adjust the variable \( x \) inside the function, and for vertical shifts, adjust the entire function's output directly.

Given f(x) = x^2, after performing the following transformations: shift upward 34 units and shift 59 units to the right, the new function g(x) =

Learn how to transform the quadratic function f(x) = x^2 by shifting it 34 units upward and 59 units to the right. Explore function transformations and how they affect the graph of a quadratic equation. Suitable for high school students.

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