Which of the following functions will have the same x-intercept(s) as its reflection about the origin?(1 point)

Question

Which of the following functions will have the same x-intercept(s) as its reflection about the origin?(1 point)

Responses

y=x2
y equals x squared

y=−12x+6

y equals negative Start Fraction 1 over 2 End Fraction x plus 6

y=(x−4)2

y equals left parenthesis x minus 4 right parenthesis squared

y=2x−5

Answer 1

Bot GPT 4-o mini answered
2 hours ago

To determine which of the given functions will have the same x-intercept(s) as its reflection about the origin, we need to understand how to find the x-intercepts of a function and how its reflection about the origin behaves.

The reflection of a function \( f(x) \) about the origin is given by \( -f(x) \).

Step 1: Find the x-intercepts of each function

1. For \( y = x^2 \):

Set \( x^2 = 0 \).
\( x = 0 \). (x-intercept is at (0, 0))

2. For \( y = -\frac{1}{2}x + 6 \):

Set \( -\frac{1}{2}x + 6 = 0 \).
\( -\frac{1}{2}x = -6 \rightarrow x = 12 \). (x-intercept is at (12, 0))

3. For \( y = (x - 4)^2 \):

Set \( (x - 4)^2 = 0 \).
\( x - 4 = 0 \rightarrow x = 4 \). (x-intercept is at (4, 0))

4. For \( y = 2x - 5 \):

Set \( 2x - 5 = 0 \).
\( 2x = 5 \rightarrow x = \frac{5}{2} \). (x-intercept is at \((\frac{5}{2}, 0)\))

Step 2: Find the reflections and their x-intercepts

1. Reflecting \( y = x^2 \):

Reflect: \( y = -x^2 \).
Set \( -x^2 = 0 \) → \( x^2 = 0 \) → \( x = 0 \). (x-intercept is (0, 0))

2. Reflecting \( y = -\frac{1}{2}x + 6 \):

Reflect: \( y = \frac{1}{2}x - 6 \).
Set \( \frac{1}{2}x - 6 = 0 \) → \( \frac{1}{2}x = 6 \rightarrow x = 12 \). (x-intercept is (12, 0))

3. Reflecting \( y = (x - 4)^2 \):

Reflect: \( y = -(x - 4)^2 \).
Set \( -(x - 4)^2 = 0 \) → \( (x - 4)^2 = 0 \rightarrow x = 4 \). (x-intercept is (4, 0))

4. Reflecting \( y = 2x - 5 \):

Reflect: \( y = -2x + 5 \).
Set \( -2x + 5 = 0 \) → \( 2x = 5 \rightarrow x = \frac{5}{2} \). (x-intercept is \((\frac{5}{2}, 0)\))

Conclusion

Now, we can compare the x-intercepts of each function and its reflection:

For \( y = x^2 \) and its reflection: Both have x-intercept at (0, 0).
For \( y = -\frac{1}{2}x + 6 \) and its reflection: Both have x-intercept at (12, 0).
For \( y = (x - 4)^2 \) and its reflection: Both have x-intercept at (4, 0).
For \( y = 2x - 5 \) and its reflection: Both have x-intercept at \((\frac{5}{2}, 0)\).

Answer

Since all provided functions have the same x-intercept as their reflection about the origin, the correct answer is:

All of the functions provided have the same x-intercepts as their reflections about the origin.