Question

Question 3: Consider the relationship between y=f(x) and y=f(-x).
Part A fill in the blanks (3 points): What is this type of function transformation called? What does it look like on a graph?
Answer: This type of transformation is called a [blank]. Specifically, it is a reflection over the [blank]-axis. On a graph, this means that every point on the graph of y=f(-x) [blank].
Part B Sketch a graph (7 points):
Choose one of the parent functions from question 1.(F(x)= 3√x
F (x) = 2^x
F (x) = x
f(x)= x^2
F (x) = √x

)create a full graph with all positive, negative, positive-negative, negative-positive areas on the graph. both the parent function and the transformation. Be sure to clearly label each sketch.

Answer 1

### Part A

This type of transformation is called a **horizontal reflection**. Specifically, it is a reflection over the **y-axis**. On a graph, this means that every point on the graph of \( y = f(-x) \) **has its x-coordinates negated, resulting in a mirror image of the graph of \( y = f(x) \) across the y-axis**.

---

### Part B

Let's choose one of the parent functions to sketch. We'll take \( f(x) = x^2 \) (the quadratic function) for this example.

1. **Sketch the Parent Function \( y = x^2 \)**:
- The graph is a parabola that opens upwards with its vertex at the origin (0,0).
- It takes values such that \( y \geq 0 \) for all \( x \).
- The points include:
- (−2, 4)
- (−1, 1)
- (0, 0)
- (1, 1)
- (2, 4)

2. **Sketch the Transformed Function \( y = f(-x) = (-x)^2 \)**:
- Since squaring a number results in the same output regardless of the sign of the input, \( y = f(-x) = (-x)^2 = x^2 \).
- Therefore, the graph of \( y = f(-x) \) looks identical to \( y = f(x) \), and it also opens upwards with a vertex at the origin (0,0).
- The points will remain the same:
- (-2, 4)
- (-1, 1)
- (0, 0)
- (1, 1)
- (2, 4)

### Graph

Here's how you could sketch these graphs:

1. **Draw the Cartesian coordinates** on the graph (x-axis and y-axis).
2. **Plot the points** for both \( y = x^2 \) and \( y = f(-x) \), noting that they overlap.
3. **Label** both curves clearly:
- Label the original function in blue: \( y = x^2 \)
- Label the transformed function in red: \( y = f(-x) = x^2 \)

### Example of what it would look like:

```
| * *
| * * * *
| * * * *
| * *
---+-------------------------
|* *
| *
| *
|___________________________
-3 -2 -1 0 1 2 3
```

In this sketch:
- The asterisks represent points on the graph of \( y = x^2 \) (and thus the same points for \( y = f(-x)\)).
- Each curve simply mirrors perfectly over the y-axis, showing that \( f(-x) \) results in no change for the quadratic function.

Make sure to name the axes and provide a key for the colors if coloring the lines for better visualization!