Graphing

Some Transformations

We can sometimes obtain the graph of a function $y=f(x)$ from the graph of a simpler one by applying some of the following transformations:

Part 1: Vertical stretch or compression

One type of transformation involves multiplying the whole function by a nonzero number. $$f(x) \longrightarrow af(x)$$ Let $g(x)=af(x)$. If $a<0$, the graph of $f(x)$ will be reflected about the $x$-axis to obtain the graph of $g(x)$. Furthermore:

If $|a|<1$ the graph of $f(x)$ will be compressed vertically to obtain $g(x)$.
If $|a|>1$ the graph of $f(x)$ will be stretched vertically to obtain $g(x)$.

Example. Draw the graph of $y=-2\sin(x)$.

Solution: We can obtain it by, starting from the graph of $y=\sin(x)$, we first reflect through the $x$-axis and then stretch the graph by a factor of two. The result is:

Part 2: Horizontal stretch or compression

We can also transform a function by multiplying the variable by a nonzero number. $$f(x) \longrightarrow f(bx)$$ Let $g(x)=f(bx)$. If $b<0$, the graph of $f(x)$ will be reflected through the $y$-axis to obtain the graph of $g(x)$. Furthermore:

If $|b|<1$ the graph of $f(x)$ will be stretched horizontally to obtain $g(x)$.
If $|b|>1$ the graph of $f(x)$ will be compressed horizontally to obtain $g(x)$.

Example. Draw the graph of $y=\tan\left(\frac{x}{3}\right)$.

Solution: We can start from the graph of $y=\tan(x)$ and stretch it by a factor of $3$.

Part 3: Vertical translation

Adding a nonzero number to a function results in translating its graph vertically: $$f(x) \longrightarrow f(x) + c$$ Let $g(x)=f(x)+c$.

If $c<0$ the graph of $f(x)$ will be shifted down $c$ units to obtain $g(x)$.
If $c>0$ the graph of $f(x)$ will be shifted up $c$ units to obtain $g(x)$.

Example. Draw the graph of $y=x^2+5$.

Solution: We translate the graph of $y=x^2$ up by $5$ units to obtain the required graph.

Part 4: Horizontal translation

Another transformation involves adding a nonzero number to the variable $x$: $$f(x) \longrightarrow f(x+d)$$ Let $g(x)=f(x+d)$.

If $d<0$ the graph of $f(x)$ will be shifted $d$ units to the right to obtain $g(x)$.
If $d>0$ the graph of $f(x)$ will be shifted $d$ units to the left to obtain $g(x)$.

Example. Draw the graph of $y=e^{x-4}$.

Solution: Starting from the graph of $y=e^x$, we shift $4$ units to the right.



We can also do a sequence of these transformations when more complicated graphs are involved.

Example. Draw the graph of $y=\frac{2}{\sqrt{4-x}}$.

Solution: Starting from the graph of $y=\frac{1}{\sqrt{x}}$, we perform vertical stretching, horizontal reflection and horizontal shift to get $y=\frac{2}{\sqrt{4-x}}$: