By applying certain transformations to the graph of a given function we can obtain the graphs of related functions. This will give us the ability to sketch the graphs of many functions quickly by hand. It will also enable us to write equations for given graphs.
Let’s first consider translations of graphs. If $c$ is a positive number, then the graph of $y = f(x) + c$ is just the graph of $y = f(x)$ shifted upward a distance of $c$ units (because each $y$-coordinate is increased by the same number $c$).
Likewise, if $g(x)= f(x-c)$, where $c > 0$, then the value of $g$ at $x$ is the same as the value of $f$ at $x - c$ ($c$ units to the left of $x$). Therefore the graph of $y= f(x-c)$ is just the graph of $y= f(x)$ shifted $c$ units to the right.
Suppose $c>0$. To obtain the graph of
$$ y=f(x)+c, \text{ shift the graph of } y=f(x) \text{ a distance } c \text{ units upward} $$
$$ y=f(x)-c, \text{ shift the graph of } y=f(x) \text{ a distance } c \text{ units downward} $$
$$ y=f(x-c), \text{ shift the graph of } y=f(x) \text{ a distance } c \text{ units to the right} $$
$$ y=f(x+c), \text{ shift the graph of } y=f(x) \text{ a distance } c \text{ units to the left} $$
Now let’s consider the stretching and reflecting transformations. If $c > 1$, then the graph of $y = cf(x)$ is the graph of $y = f(x)$ stretched by a factor of $c$ in the vertical direction (because each $y$-coordinate is multiplied by the same number $c$). The graph of $y = -f(x)$ is the graph of $y = f(x)$ reflected about the $x$-axis because the point $(x, y)$ is replaced by the point $(x, 2y)$.
For instance, in order to get the graph of $y = 2\cos(x)$ we multiply the $y$-coordinate of each point on the graph of $y = \cos(x)$ by $2$. This means that the graph of $y = \cos(x)$ gets stretched vertically by a factor of $2$.
Suppose $c > 1$. To obtain the graph of
$$ y=cf(x), \text{ stretch the graph of } y=f(x) \text{ vertically by a factor of } c $$
$$ y=\left(\frac{1}{c}\right)f(x), \text{ shrink the graph of } y=f(x) \text{ vertically by a factor of } c $$
$$ y=f(cx), \text{ shrink the graph of } y=f(x) \text{ horizontally by a factor of } c $$
$$ y=f\left(\frac{x}{c}\right), \text{ stretch the graph of } y=f(x) \text{ horizontally by a factor of } c $$