Reflect curves and asymptotes

Reflect coordinates and asymptotes

A reflection changes the sign of one coordinate. The named axis stays fixed: reflecting in the x-axis changes y; reflecting in the y-axis changes x.

Reflect in the x-axis

Replace y by $-y$. Then $-y=f(x)$ gives $y=-f(x)$, with point map $(p,q)\mapsto(p,-q)$.

Every point moves to the same distance on the opposite side of the x-axis. A minimum becomes a maximum. This is signed scaling by $-1$ parallel to the y-axis.

Reflect in the y-axis

Replace x by $-x$ to get $y=f(-x)$, with point map $(p,q)\mapsto(-p,q)$.

The height stays the same while the horizontal position reverses. This is signed scaling by $-1$ parallel to the x-axis.

Reflect asymptotes too

• In Example 5, the horizontal asymptote $y=1$ becomes $-y=1$, so $y=-1$. The vertical asymptote $x=-1$ stays fixed because its equation does not contain y.
• In Example 6, $x=-1$ becomes $-x=-1$, so the vertical asymptote is $x=1$. The horizontal asymptote $y=1$ stays fixed because its equation does not contain x.

Each example asks for separate graphs with both branches. In Self-Practice 3, compare $-f(x)$ and $f(-x)$ for $f(x)=x+1$: both lines have gradient $-1$, but their intercepts differ.

Check before the examples: $(2,-3)$ becomes $(2,3)$ under an x-axis reflection and $(-2,-3)$ under a y-axis reflection. The coordinate along the mirror axis does not change.

Reflection activity and source notes

The two quadratic activities include the Teacher’s completed stationary-point and intercept tables. Self-Practice 3’s solutions are independently derived; neither source copy prints answers on this page.