Q: What is transformations of tangent?

For $y = a \tan(b(x - h)) + d$:

Q: What is tangent with horizontal compression?

$y = \tan(3 x)$: period $\frac{\pi}{3}$. Asymptotes at $x = \frac{\pi}{6} + \frac{k \pi}{3}$, that is at $\frac{\pi}{6}, \frac{\pi}{2}, \frac{5 \pi}{6}, \dots$. Zeros at $x = \frac{k \pi}{3}$.

Question 1

What is transformations of sine and cosine?

Accepted Answer

For $y = a \sin(b(x - h)) + d$ or $y = a \cos(b(x - h)) + d$:

Question 2

What is transformations of tangent?

Accepted Answer

For $y = a 	an(b(x - h)) + d$:

Question 3

What is reading features off the equation?

Accepted Answer

Given any equation in the standard form, you can extract amplitude, period and shifts in seconds without sketching. Reverse process: given amplitude, period, centre line and a starting point, write the equation.

Question 4

What is amplitude, period and centre?

Accepted Answer

$y = 4 \sin(\pi x) + 2$: amplitude $4$, period $\frac{2 \pi}{\pi} = 2$, centre line $y = 2$. Max $6$, min $-2$.

Question 5

What is phase shift in the standard form?

Accepted Answer

$y = \sin\left( 2 x + \frac{\pi}{2} \right) = \sin\left( 2\left( x + \frac{\pi}{4} \right) \right)$. Phase shift: left by $\frac{\pi}{4}$. Period $\pi$.

Question 6

What is tangent with horizontal compression?

Accepted Answer

$y = 	an(3 x)$: period $\frac{\pi}{3}$. Asymptotes at $x = \frac{\pi}{6} + \frac{k \pi}{3}$, that is at $\frac{\pi}{6}, \frac{\pi}{2}, \frac{5 \pi}{6}, \dots$. Zeros at $x = \frac{k \pi}{3}$.

Question 7

What is writing an equation from features?

Accepted Answer

Find a cosine equation with amplitude $5$, period $\frac{\pi}{2}$, centre line $y = 1$, and the first maximum at $x = \frac{\pi}{8}$.

Question 8

What is a reflected sine?

Accepted Answer

$y = -3 \sin x + 1$: amplitude $3$, period $2 \pi$, centre line $y = 1$. The negative coefficient reflects: the graph starts at the centre line at $x = 0$ and goes down to the minimum at $y = -2$ first, instead of up to the maximum.

Question 9

What is confusing period with $b$?

Accepted Answer

$b$ is not the period; the period is $\frac{2 \pi}{|b|}$ (or $\frac{\pi}{|b|}$ for tan).

Question 10

What is mis-reading the phase shift?

Accepted Answer

In $y = \sin(b x + c)$, the phase shift is $-\frac{c}{b}$, not $-c$. Factor $b$ out of the bracket first.

Question 11

What is confusing amplitude with maximum value?

Accepted Answer

Amplitude is the half-distance from minimum to maximum. The maximum value is $d + |a|$, not just $|a|$.

Question 12

What is forgetting tangent's period is $\pi$?

Accepted Answer

Sine and cosine have period $2 \pi$; tangent has period $\pi$. Use the right formula.

Question 13

What is dropping the absolute value in amplitude?

Accepted Answer

Amplitude is $|a|$, always non-negative. A negative $a$ flips the curve but the amplitude is still $|a|$.