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

Mar 29, 2016

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

#### Explanation:

We will construct this graph in the following sequence of steps:

1. $y = \cos \left(x\right)$
graph{cos(x) [-5, 5, -3, 3]}

2. $y = \cos \left(3 x\right)$
This transformation squeezes the graph horizontally along the X-axis towards Y-axis by a factor of $3$ because, if point $\left(a , b\right)$ belongs to graph of $y = f \left(x\right)$ (that is, $a$ and $b$ satisfy $b = f \left(a\right)$ equation) then point $\left(\frac{a}{K} , b\right)$ belongs to graph $y = f \left(K x\right)$ since $f \left(K \frac{a}{K}\right) = f \left(a\right) = b$
graph{cos(3(x)) [-5, 5, -3, 3]}

3. $y = \frac{3}{4} \cos \left(3 x\right)$
This transformation stretches the graph vertically along the Y-axis by a factor of $\frac{3}{4}$ because, if point $\left(a , b\right)$ belongs to graph of $y = f \left(x\right)$ (that is, $a$ and $b$ satisfy $b = f \left(a\right)$ equation) then point $\left(a , K b\right)$ belongs to graph $y = K f \left(x\right)$ since $K f \left(a\right) = K b$
graph{3/4cos(3(x)) [-5, 5, -3, 3]}

4. $y = \frac{3}{4} \cos \left(3 \left(x - 2\right)\right)$
This transformation shifts the graph horizontally along the X-axis by $2$ to the right because, if point $\left(a , b\right)$ belongs to graph of $y = f \left(x\right)$ (that is, $a$ and $b$ satisfy $b = f \left(a\right)$ equation) then point $\left(a + \delta , b\right)$ belongs to graph $y = f \left(x - \delta\right)$ since $f \left(a + \delta - \delta\right) = f \left(a\right) = b$
graph{3/4cos(3(x-2)) [-5, 5, -3, 3]}

5. $y = \frac{3}{4} \cos \left(3 \left(x - 2\right)\right) - 1$
This transformation shifts the graph vertically along the Y-axis by $1$ down because, if point $\left(a , b\right)$ belongs to graph of $y = f \left(x\right)$ (that is, $a$ and $b$ satisfy $b = f \left(a\right)$ equation) then point $\left(a , b - \delta\right)$ belongs to graph $y = f \left(x\right) - \delta$ since $f \left(a\right) - \delta = b - \delta$
graph{3/4cos(3(x-2))-1 [-5, 5, -3, 3]}