# How do you graph  y=4cos(2x-1)+3?

Apr 1, 2018

So features of graph

Amp:4
Period: $\pi$
Max:7
Min:-1

graph{4cos(2x-1)+3 [-10, 10, -5, 5]}

#### Explanation:

Translation form of sine is

$A \sin \left[B \left(x - C\right)\right] + D$

A~ Vertical stretch/Amp, y values get stretched by A
B~ Horizontal strech/Period, x values get stretched by $\frac{1}{B}$
C~ Horizontal translation/Phase shift, x values move over by C
D~ Vertical translation, y values up by D

So we know that if we put it in translation form it looks like

$4 \cos \left[2 \left(x - \frac{1}{2}\right)\right] + 3$

So we know that the original $\sin \left(x\right)$ has these features

Amp:1
Period:$2 \pi$
Max: 1
Min: -1

graph{sin(x) [-8.89, 8.886, -4.446, 4.446]}

So $4 \cos \left(x\right)$ means the Amplitude becomes 4 meaning the max is 4 and min -4

So $4 \cos \left(2 x\right)$ means the period halves becoming $\pi$

So $4 \cos \left[2 \left(x - \frac{1}{2}\right)\right]$ means the origin moves over by $\frac{1}{2}$

$4 \cos \left[2 \left(x - \frac{1}{2}\right)\right] + 3$ means all the y values move up by 3 meaning the max is 7 and the min -1

So features of graph

Amp:4
Period: $\pi$
Max:7
Min:-1

graph{4cos(2x-1)+3 [-10, 10, -5, 5]}