$y = f(x - h)$ shifts right by $h$ (left if $h < 0$). Note the sign convention: $(x - h)$ means shift right.

$y = a f(x)$ stretches vertically by factor $a$ (compresses if $0 < a < 1$, reflects if $a < 0$).

$y = f(b x)$ stretches horizontally by factor $1/b$ (compresses if $b > 1$).

Question 1

What is example 1. Linear?

Accepted Answer

$y = 2x - 3$. Gradient 2, $y$-intercept $-3$. $x$-intercept: $0 = 2x - 3$, $x = 1.5$. Sketch as straight line through $(0, -3)$ and $(1.5, 0)$.

Question 2

What is example 2. Quadratic transformation?

Accepted Answer

Start with $y = x^2$ (parabola, vertex at origin). Apply $y = 2(x - 1)^2 - 5$. This is:

Question 3

What is example 3. Exponential transformation?

Accepted Answer

$y = 2 \cdot 3^x + 1$. Start with $y = 3^x$. Apply dilation by 2 (stretches vertically), then translation up by 1. New horizontal asymptote: $y = 1$.

Question 4

What is translation in $y$?

Accepted Answer

$y = f(x) + k$ shifts up by $k$ (down if $k < 0$).

Question 5

What is translation in $x$?

Accepted Answer

$y = f(x - h)$ shifts right by $h$ (left if $h < 0$). Note the sign convention: $(x - h)$ means shift right.

Question 6

What is dilation in $y$?

Accepted Answer

$y = a f(x)$ stretches vertically by factor $a$ (compresses if $0 < a < 1$, reflects if $a < 0$).

Question 7

What is dilation in $x$?

Accepted Answer

$y = f(b x)$ stretches horizontally by factor $1/b$ (compresses if $b > 1$).

Question 8

What is combined transformations?

Accepted Answer

$y = a f(b(x - h)) + k$ combines all four with vertex at $(h, k)$.

Question 9

What is reflections?

Accepted Answer

Special case of dilations:

Question 10

What is translation sign error?

Accepted Answer

$y = f(x - 3)$ shifts right by 3 (positive direction), not left.

Question 11

What is wrong order of transformations?

Accepted Answer

Apply inside-the-bracket transformations first (operations on $x$), then outside (operations on $y$).

Question 12

What is forgetting the asymptote on exponential / log graphs?

Accepted Answer

Exponentials have horizontal asymptotes; logs have vertical asymptotes. Mark them.

Question 13

What is confusing domain with range?

Accepted Answer

Domain is the set of valid inputs; range is the set of outputs. For logs, the domain is restricted to positive $x$.