Translate curves and asymptotes

Inside changes act on the input before the function is evaluated. Use a labelled point to verify horizontal scale and direction. Apply absolute-value folds only to the relevant parts of the graph.

Translate the whole curve, including its asymptotes

A translation moves every point the same distance and direction. The curve keeps its shape and size. Move its asymptotes too: they belong to the graph, even though they are not part of the curve.

For the rules below, $a>0$ and $(p,q)$ is a point on $y=f(x)$.

Move right or left

• Right by a: replace x by $x-a$, giving $y=f(x-a)$. The point moves by $(p,q)\mapsto(p+a,q)$.
• Left by a: replace x by $x+a$, giving $y=f(x+a)$. The point moves by $(p,q)\mapsto(p-a,q)$.

Why does a minus sign move the graph right? To get the old output $f(p)$ from $f(x-a)$, solve $x-a=p$. The new input is $x=p+a$.

Move up or down

• Up by a: replace y by $y-a$. Then $y-a=f(x)$ gives $y=f(x)+a$, and $(p,q)\mapsto(p,q+a)$.
• Down by a: replace y by $y+a$. Then $y+a=f(x)$ gives $y=f(x)-a$, and $(p,q)\mapsto(p,q-a)$.

Move the asymptotes with the curve

In Example 1, replace x by $x-1$ in the vertical asymptote: $x=0$ becomes $x-1=0$, so the new asymptote is $x=1$. The horizontal asymptote $y=0$ stays fixed.

In Example 2, replace y by $y-1$: the asymptote $y=0$ becomes $y-1=0$, or $y=1$. The vertical asymptote $x=0$ stays fixed. Draw both branches on separate axes, as each question requests.

Check before the examples: a point $(2,3)$ lies on $y=f(x)$. On $y=f(x-1)+2$, it moves to $(3,5)$: one unit right and two up.

Activities and source notes

The opening calculator activities compare three parabolas in each direction. Use their minimum points to check that every point moves by the same amount. The worked tables include the Teacher’s completed minima and labels. Self-Practice 1 asks for two separate straight-line sketches; its worked answers are independently derived because neither source copy prints them here.

The printed Explore transformations! QR resource links to the source exploration. Its current destination content has not been reviewed here.

The later units cover scaling, reflection, combinations, $|f(x)|$, $f(|x|)$ and $1/f(x)$. The source references are Steward, J. (2015), Calculus: Early Transcendentals, 8th ed., Cengage Learning, and Burghes, Cassell, Cross, Deft, Middle and Thallon (1994), Pure Mathematics, Heinemann Educational. The original bibliography spelling is retained in the native transcription.