Question
Graph the function f(x)=(x+1)^2
and the function of its horizontal reflection. What is the position of the point (−4,9)
in the reflected function?(1 point)
and the function of its horizontal reflection. What is the position of the point (−4,9)
in the reflected function?(1 point)
Answers
Answered by
GPT-4o mini
To graph the function \( f(x) = (x + 1)^2 \), let's first examine its basic properties.
1. **Original Function \( f(x) = (x + 1)^2 \)**:
- This is a standard parabola that opens upwards.
- The vertex of the parabola is at the point \((-1, 0)\) since the equation is in the form \( (x - h)^2 + k \) where \((h, k)\) is the vertex.
- The y-intercept occurs when \( x = 0 \):
\[
f(0) = (0 + 1)^2 = 1^2 = 1
\]
- The point \( (-4, 9) \) can be calculated by substituting \( x = -4 \):
\[
f(-4) = (-4 + 1)^2 = (-3)^2 = 9
\]
Therefore, \( (-4, 9) \) is indeed on the original curve.
2. **Horizontal Reflection**:
- A horizontal reflection of the function occurs by replacing \( x \) with \( -x \). Thus, we get the reflected function \( g(x) = f(-x) = ((-x) + 1)^2 = (1 - x)^2 \).
3. **Analyzing the Reflected Function** \( g(x) = (1 - x)^2 \):
- This is also a parabola, and it opens upwards.
- The vertex of this parabola is at the point \( (1, 0) \).
- To understand the position of the point \( (-4, 9) \) in this reflected function, we need to find its corresponding point in \( g(x) \).
4. **Finding the Corresponding Point**:
- For the original function, when \( f(x) = 9 \), we already calculated this occurs at \( x = -4 \) (since \( f(-4) = 9 \)).
- Now we will find \( g(-4) \):
\[
g(-4) = (1 - (-4))^2 = (1 + 4)^2 = 5^2 = 25
\]
Therefore, the point \( (-4, 9) \) in the original function corresponds to the point \( (-4, 25) \) in the reflected function.
### Conclusion:
The position of the point \( (-4, 9) \) in the reflected function \( g(x) = (1 - x)^2 \) is at \( (-4, 25) \).
1. **Original Function \( f(x) = (x + 1)^2 \)**:
- This is a standard parabola that opens upwards.
- The vertex of the parabola is at the point \((-1, 0)\) since the equation is in the form \( (x - h)^2 + k \) where \((h, k)\) is the vertex.
- The y-intercept occurs when \( x = 0 \):
\[
f(0) = (0 + 1)^2 = 1^2 = 1
\]
- The point \( (-4, 9) \) can be calculated by substituting \( x = -4 \):
\[
f(-4) = (-4 + 1)^2 = (-3)^2 = 9
\]
Therefore, \( (-4, 9) \) is indeed on the original curve.
2. **Horizontal Reflection**:
- A horizontal reflection of the function occurs by replacing \( x \) with \( -x \). Thus, we get the reflected function \( g(x) = f(-x) = ((-x) + 1)^2 = (1 - x)^2 \).
3. **Analyzing the Reflected Function** \( g(x) = (1 - x)^2 \):
- This is also a parabola, and it opens upwards.
- The vertex of this parabola is at the point \( (1, 0) \).
- To understand the position of the point \( (-4, 9) \) in this reflected function, we need to find its corresponding point in \( g(x) \).
4. **Finding the Corresponding Point**:
- For the original function, when \( f(x) = 9 \), we already calculated this occurs at \( x = -4 \) (since \( f(-4) = 9 \)).
- Now we will find \( g(-4) \):
\[
g(-4) = (1 - (-4))^2 = (1 + 4)^2 = 5^2 = 25
\]
Therefore, the point \( (-4, 9) \) in the original function corresponds to the point \( (-4, 25) \) in the reflected function.
### Conclusion:
The position of the point \( (-4, 9) \) in the reflected function \( g(x) = (1 - x)^2 \) is at \( (-4, 25) \).