Question
Consider the relationship between y = f(x) and y = f (-х).
Question 1: What is this type of function transformation called? What does it look like on a graph?
Question 2: The parent function is f(x) = x^2. On a sheet of graph
paper, sketch both y = f(x) and y = f (-x) for the parent
function. Be sure to clearly label each sketch.
Question 1: What is this type of function transformation called? What does it look like on a graph?
Question 2: The parent function is f(x) = x^2. On a sheet of graph
paper, sketch both y = f(x) and y = f (-x) for the parent
function. Be sure to clearly label each sketch.
Answers
GPT-4o mini
### Question 1: Function Transformation
The transformation between \( y = f(x) \) and \( y = f(-x) \) is known as a **horizontal reflection** or **horizontal flip**.
On a graph, this transformation reflects the graph of the function \( f(x) \) across the **y-axis**. If a function \( f(x) \) is symmetric about the y-axis, which means it is an **even function** (such that \( f(x) = f(-x) \)), then the graph will look the same after the transformation. If the function is not even, the shape will change and will be mirrored about the y-axis.
### Question 2: Sketching the Parent Function and Its Reflection
For the parent function \( f(x) = x^2 \):
- **Original function:** The graph of \( f(x) = x^2 \) is a parabola opening upwards, with its vertex at the origin (0,0). The graph is symmetric about the y-axis.
- **Reflected function:** The graph of \( f(-x) = (-x)^2 = x^2 \) will look the same as \( f(x) \) because squaring a negative, as well as a positive number, yields the same result (since it's an even function).
### Sketches:
1. **Sketch of \( y = f(x) = x^2 \)**:
\[
\begin{array}{c}
\text{(0, 0)} \\
\text{*} \\
\text{* *} \\
\text{* *} \\
\text{* *} \\
\text{* *} \\
\text{(X-axis)} \\
\end{array}
\]
(Parabola opens upward, vertex at (0,0), symmetric about the y-axis.)
2. **Sketch of \( y = f(-x) = x^2 \)**:
Since the function is \( y = x^2 \), the sketch will look exactly the same as the original function:
\[
\begin{array}{c}
\text{(0, 0)} \\
\text{*} \\
\text{* *} \\
\text{* *} \\
\text{* *} \\
\text{* *} \\
\text{(X-axis)} \\
\end{array}
\]
(It is also a parabola that opens upward and appears identical to the first.)
### Labels:
- Label the first sketch as \( y = f(x) = x^2 \).
- Label the second sketch as \( y = f(-x) = x^2 \).
Both sketches will look identical because the parent function \( x^2 \) is an even function.
The transformation between \( y = f(x) \) and \( y = f(-x) \) is known as a **horizontal reflection** or **horizontal flip**.
On a graph, this transformation reflects the graph of the function \( f(x) \) across the **y-axis**. If a function \( f(x) \) is symmetric about the y-axis, which means it is an **even function** (such that \( f(x) = f(-x) \)), then the graph will look the same after the transformation. If the function is not even, the shape will change and will be mirrored about the y-axis.
### Question 2: Sketching the Parent Function and Its Reflection
For the parent function \( f(x) = x^2 \):
- **Original function:** The graph of \( f(x) = x^2 \) is a parabola opening upwards, with its vertex at the origin (0,0). The graph is symmetric about the y-axis.
- **Reflected function:** The graph of \( f(-x) = (-x)^2 = x^2 \) will look the same as \( f(x) \) because squaring a negative, as well as a positive number, yields the same result (since it's an even function).
### Sketches:
1. **Sketch of \( y = f(x) = x^2 \)**:
\[
\begin{array}{c}
\text{(0, 0)} \\
\text{*} \\
\text{* *} \\
\text{* *} \\
\text{* *} \\
\text{* *} \\
\text{(X-axis)} \\
\end{array}
\]
(Parabola opens upward, vertex at (0,0), symmetric about the y-axis.)
2. **Sketch of \( y = f(-x) = x^2 \)**:
Since the function is \( y = x^2 \), the sketch will look exactly the same as the original function:
\[
\begin{array}{c}
\text{(0, 0)} \\
\text{*} \\
\text{* *} \\
\text{* *} \\
\text{* *} \\
\text{* *} \\
\text{(X-axis)} \\
\end{array}
\]
(It is also a parabola that opens upward and appears identical to the first.)
### Labels:
- Label the first sketch as \( y = f(x) = x^2 \).
- Label the second sketch as \( y = f(-x) = x^2 \).
Both sketches will look identical because the parent function \( x^2 \) is an even function.