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

Question

y = 2x - 5
y = x^2
y = (x - 4)^2
y = -1/2x + 6

Answer 1

To determine which of the given functions will have the same x-intercept(s) as its reflection about the origin, we need to first understand what the reflection of a function about the origin means.

When a function $y=f(x)$ is reflected about the origin, it transforms into $y=-f(-x)$. The x-intercepts of a function are the points where $y=0$. Therefore, we need to find the x-intercepts of both the original function and its reflection, and see if they are the same.

Let's analyze each function one by one.

1. Function: $y=2x-5$

   • To find the x-intercept, set $y=0$: $$0=2x-5⟹2x=5⟹x=\frac{5}{2}$$
   • Reflecting the function yields $y=-(2(-x)-5)=-(-2x-5)=2x+5$.
   • To find the x-intercept of the reflection: $$0=2x+5⟹2x=-5⟹x=-\frac{5}{2}$$
   • The x-intercepts are different.

2. Function: $y=x^2$

   • Finding the x-intercept: $$0=x^2⟹x=0$$
   • Reflecting the function yields $y=-(-x{)}^2=-x^2$.
   • The x-intercept of the reflection: $$0=-x^2⟹x^2=0⟹x=0$$
   • The x-intercepts are the same.

3. Function: $y=(x-4{)}^2$

   • Finding the x-intercept: $$0=(x-4{)}^2⟹x-4=0⟹x=4$$
   • Reflecting the function yields $y=-((-x)-4{)}^2=-(-x-4{)}^2$.
   • The x-intercept of the reflection: $$0=-(x+4{)}^2⟹(x+4{)}^2=0⟹x=-4$$
   • The x-intercepts are different.

4. Function: $y=-\frac{1}{2}x+6$

   • Finding the x-intercept: $$0=-\frac{1}{2}x+6⟹\frac{1}{2}x=6⟹x=12$$
   • Reflecting the function yields $y=-(-\frac{1}{2}(-x)+6)=-(\frac{1}{2}x+6)=-\frac{1}{2}x-6$.
   • The x-intercept of the reflection: $$0=-\frac{1}{2}x-6⟹-\frac{1}{2}x=6⟹x=-12$$
   • The x-intercepts are different.

In summary, the only function that maintains the same x-intercept as its reflection about the origin is:

$y=x^2$.