New Zealand
Level 8 - NCEA Level 3

# Transformations of sine and cosine curves and equations

## Interactive practice questions

The functions $f\left(x\right)$f(x) and $g\left(x\right)=f\left(x-k\right)-j$g(x)=f(xk)j have been graphed on the same set of axes in grey and black respectively.

a

What transformations have occurred from $f\left(x\right)$f(x) to $g\left(x\right)$g(x)?

Select all that apply.

Horizontal translation of $\frac{\pi}{3}$π3 radians right.

A

Vertical translation of $\frac{\pi}{3}$π3 units up.

B

Vertical translation of $3$3 units down.

C

Horizontal translation of $3$3 radians left.

D

Horizontal translation of $\frac{\pi}{3}$π3 radians right.

A

Vertical translation of $\frac{\pi}{3}$π3 units up.

B

Vertical translation of $3$3 units down.

C

Horizontal translation of $3$3 radians left.

D
b

Determine the value of $j$j.

c

Determine the smallest positive value of $k$k.

Easy
Approx 2 minutes

The graph of $y=\cos x$y=cosx has been transformed into the graph of $y=\cos\left(2x+\frac{\pi}{3}\right)$y=cos(2x+π3).

Consider the graphs of $y=\sin x$y=sinx and $y=5\sin\left(x+\frac{\pi}{4}\right)$y=5sin(x+π4).

Consider the graphs of $y=\cos x$y=cosx and $y=3\cos\left(x-\frac{\pi}{4}\right)$y=3cos(xπ4).

### Outcomes

#### M8-2

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions