Graphing Related Functions

In short: Related graphs are built from a known graph $y=f(x)$. For $y=\frac{1}{f(x)}$, the zeros of $f$ become vertical asymptotes and big heights become small. For $y=|f(x)|$, every part below the x-axis flips up, and for $y=f(|x|)$, the right half is mirrored in the y-axis.

Starting point: what you already know

You will use these ideas from Years 9 and 10 and from Mathematics Advanced.

• Function notation. $f(x)$ is the height of the graph above the point $x$. So $f(3)=5$ means the point $(3,5)$ is on the graph.
• The hyperbola $y=\frac{1}{x}$ has a vertical asymptote at $x=0$ and a horizontal asymptote at $y=0$. An asymptote is a line the graph gets closer and closer to but never reaches.
• Absolute value. $|x|$ is the distance of $x$ from 0, so it is never negative. $|5|=5$ and $|-5|=5$.
• The graph of $y=|x|$ is a V shape with its corner at the origin.

The reciprocal: y = 1/f(x)

To get $y=\frac{1}{f(x)}$, take every height of $f$ and turn it upside down: a height of 2 becomes $\frac{1}{2}$, and a height of $-4$ becomes $-\frac{1}{4}$. Use these rules.

1. Zeros become asymptotes. Where $f(x)=0$, you would divide by zero, so $\frac{1}{f(x)}$ has a vertical asymptote.
2. Big becomes small. Where $f(x)$ is huge, $\frac{1}{f(x)}$ is close to 0.
3. 1 and $-1$ stay put. If $f(x)=1$ then $\frac{1}{f(x)}=1$ too. The graphs meet where the height is 1 or $-1$.
4. Same sign. Above the axis stays above. Below stays below.
5. Up becomes down. Where $f$ goes up, $\frac{1}{f}$ goes down. A minimum of $f$ becomes a maximum of $\frac{1}{f}$.

The absolute value: y = |f(x)|

Absolute value makes every height positive. So keep the parts above the x-axis and flip the parts below the axis up, like a mirror on the x-axis.

Mirror the right side: y = f(|x|)

Now the absolute value is on the $x$, not on the height. Since $f(|-3|)=f(3)$, the left side is a copy of the right side.

Rule: draw $y=f(x)$ for $x\ge 0$ only, then reflect that part in the y-axis. The result is always symmetrical about the y-axis.

Adding and subtracting graphs

For $y=f(x)+g(x)$, add the two heights at each $x$. For $y=f(x)-g(x)$, subtract them.

Useful checks:

• Where $g(x)=0$, the sum $f+g$ sits exactly on $f$.
• Where $f(x)=-g(x)$, the sum is zero.
• The sum only exists where both functions exist. The domain is the overlap of the two domains.

Quick summary

• $y=\frac{1}{f(x)}$: zeros of $f$ become vertical asymptotes, large heights become small, and the graphs meet at heights $\pm 1$.
• $y=|f(x)|$: flip every part below the x-axis up.
• $y=f(|x|)$: keep the part where $x\ge 0$ and mirror it in the y-axis.
• $y=f(x)\pm g(x)$: add or subtract heights. The domain is where both exist.

Quiz