Transformations Practice

These questions follow Transformations of Functions — the recipe $y=af(k(x-d))+c$ applied to the parent functions. Sketch on paper first; Using Desmos is for checking, after you have committed to an answer.

Describe the moves

1. Describe how the graph of $y=f(x-3)+2$ relates to the graph of $y=f(x)$.
2. Describe the transformations in $y=-2f(x)$.
3. Describe the transformations in $y=f(2x)-5$.

Answer 1

Slide right 3, slide up 2. The bracket lies about direction — $x-3$ moves right — and the outside term tells the truth, as always.

Answer 2

Vertical stretch by a factor of 2, plus a reflection in the $x$-axis. Both act on outputs: every $y$-value is doubled and flipped.

Answer 3

Horizontal compression by a factor of $\frac{1}{2}$ — inside moves run backwards, so multiplying $x$ by 2 squeezes — then a slide down 5.

Analyse and sketch

4. Starting from $y=\sqrt{x}$, describe the transformations that produce $y=\sqrt{x+1}-4$, then state its domain and range.
5. For $y=-(x-2{)}^2+3$: vertex, opening direction, domain, and range.
6. For $y=\frac{1}{x-3}+2$: describe the transformations and state the equations of both asymptotes.

Answer 4

Left 1, down 4. The domain slides with the graph: $x\ge -1$. The range slides too: $y\ge -4$. Sketch check: the “corner” of the root curve now sits at $(-1,-4)$.

Answer 5

Vertex $(2,3)$, opening down (the negative $a$). Domain: all real numbers; range: $y\le 3$ — the vertex is now a maximum, so the range hangs below it.

Answer 6

The parent $\frac{1}{x}$ slid right 3 and up 2. Its asymptotes travel with it: vertical $x=3$, horizontal $y=2$. (Hence domain $x\ne 3$ and range $y\ne 2$ — asymptotes and restrictions are the same facts in different costumes.)

Points and backwards questions

7. The point $(4,2)$ lies on $y=f(x)$. Find its image on (a) $y=3f(x-1)+2$ and (b) $y=f(2x)$.
8. Write the equation of the function obtained when $y=\sqrt{x}$ is stretched vertically by a factor of 2, reflected in the $x$-axis, then translated right 5 and up 3.

Answer 7

(a) Outputs: $y=3(2)+2=8$; inputs: $x=4+1=5$. Image: $(5,8)$. (b) Inside moves run backwards, so the $x$-coordinate divides by 2: image $(2,2)$.

Answer 8

$y=-2\sqrt{x-5}+3$. The order given — stretch and reflect first, slide second — is exactly the order the recipe applies, so each instruction maps to one letter: $a=-2$, $d=5$, $c=3$. Check one point: the parent’s $(4,2)$ should land at $(9,-1)$, and substituting $x=9$ gives $-2(2)+3=-1$. ✓