Question

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

Answer 1

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

`Asin[B(x-C)]+D`

A~ Vertical stretch/Amp, y values get stretched by A
B~ Horizontal strech/Period, x values get stretched by `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

`4cos[2(x-1/2)]+3`

So we know that the original `sin(x)` has these features

Amp:1
Period:`2pi`
Max: 1
Min: -1

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

So `4cos(x)` means the Amplitude becomes 4 meaning the max is 4 and min -4

So `4cos(2x)` means the period halves becoming `pi`

So `4cos[2(x-1/2)]` means the origin moves over by `1/2`

`4cos[2(x-1/2)]+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]}