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If the function y=x2−4 is reflected about the origin, which of the following is true regarding the number of x-intercepts of th...Question
If the function y=x2−4
is reflected about the origin, which of the following is true regarding the number of x-intercepts of the reflected function?
Hint: Sketch a graph of the function y=x2−4
, then sketch a graph of the function reflected about the origin.
Option #1: 0 x-intercepts
Option #2: 1 x-intercept
Option #3: 2 x-intercepts
(1 point)
Option #
Remaining Attempts : 3
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is reflected about the origin, which of the following is true regarding the number of x-intercepts of the reflected function?
Hint: Sketch a graph of the function y=x2−4
, then sketch a graph of the function reflected about the origin.
Option #1: 0 x-intercepts
Option #2: 1 x-intercept
Option #3: 2 x-intercepts
(1 point)
Option #
Remaining Attempts : 3
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page 7 of 8
Answers
Answered by
GPT-4o mini
To analyze the problem, let's start with the function given:
\[ y = x^2 - 4 \]
### Step 1: Find the x-intercepts of the original function
To find the x-intercepts, we set \( y = 0 \):
\[ 0 = x^2 - 4 \]
This simplifies to:
\[ x^2 = 4 \]
\[ x = \pm 2 \]
Thus, the original function \( y = x^2 - 4 \) has **2 x-intercepts** at \( x = 2 \) and \( x = -2 \).
### Step 2: Reflect the function about the origin
When reflecting a function about the origin, we replace \( (x, y) \) with \( (-x, -y) \). Therefore, the reflected function is:
\[ y = -(x^2 - 4) \]
\[ y = -x^2 + 4 \]
This is equivalent to:
\[ y = -x^2 + 4 \]
### Step 3: Find the x-intercepts of the reflected function
To find the x-intercepts of \( y = -x^2 + 4 \), we set \( y \) to 0:
\[ 0 = -x^2 + 4 \]
This simplifies to:
\[ x^2 = 4 \]
\[ x = \pm 2 \]
Thus, the reflected function \( y = -x^2 + 4 \) also has **2 x-intercepts** at \( x = 2 \) and \( x = -2 \).
### Conclusion
Both the original function and the reflected function have 2 x-intercepts. Therefore, the correct answer to the question regarding the number of x-intercepts of the reflected function is:
**Option #3: 2 x-intercepts**.
\[ y = x^2 - 4 \]
### Step 1: Find the x-intercepts of the original function
To find the x-intercepts, we set \( y = 0 \):
\[ 0 = x^2 - 4 \]
This simplifies to:
\[ x^2 = 4 \]
\[ x = \pm 2 \]
Thus, the original function \( y = x^2 - 4 \) has **2 x-intercepts** at \( x = 2 \) and \( x = -2 \).
### Step 2: Reflect the function about the origin
When reflecting a function about the origin, we replace \( (x, y) \) with \( (-x, -y) \). Therefore, the reflected function is:
\[ y = -(x^2 - 4) \]
\[ y = -x^2 + 4 \]
This is equivalent to:
\[ y = -x^2 + 4 \]
### Step 3: Find the x-intercepts of the reflected function
To find the x-intercepts of \( y = -x^2 + 4 \), we set \( y \) to 0:
\[ 0 = -x^2 + 4 \]
This simplifies to:
\[ x^2 = 4 \]
\[ x = \pm 2 \]
Thus, the reflected function \( y = -x^2 + 4 \) also has **2 x-intercepts** at \( x = 2 \) and \( x = -2 \).
### Conclusion
Both the original function and the reflected function have 2 x-intercepts. Therefore, the correct answer to the question regarding the number of x-intercepts of the reflected function is:
**Option #3: 2 x-intercepts**.
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