# How do you find the function rule?

Apr 5, 2015

You need to know some base graphs:

• $f \left(x\right) = c$
• $f \left(x\right) = x$
• $f \left(x\right) = {x}^{2}$
• $f \left(x\right) = {x}^{3}$
• $f \left(x\right) = \ln \left(x\right)$
• $f \left(x\right) = \left\mid x \right\mid$

Note: Also know these graphs' negative versions. (They are symmetric to the $x$-axis)

Look at the graph and decide which base graph is similar to the graph.

Usually, the given graph is a shifted version of a base graph.

Example:

graph{y=(x+1)^2+2 [-5, 5, -5, 5]}

This graph is similar to $f \left(x\right) = {x}^{2}$ Pick a point on the base graph. Lets say $O \left(0 , 0\right)$

The point $O$ is shifted $- 2$ units on the $x$-axis and $+ 2$ units on the $y$-axis.

When we add $+ 2$ to $x$ variable ($f \left(x\right) = {\left(x + 2\right)}^{2}$), base graph will shift $- 2$ units on the $x$-axis. (In ${x}^{2}$ $y$ was $0$ when $x$ was $0$. But in ${\left(x + 2\right)}^{2}$, $y$ is $0$ when $x$ is $- 2$)

Finally, the graph is shifted $+ 2$ in $y$-axis. Since $y = f \left(x\right)$:

$y = f \left(x\right) + 2$

$y = \left[{\left(x + 2\right)}^{2}\right] + 2 = {\left(x + 2\right)}^{2} + 2$

In my opinion, "guessing" the function rule from its graph is not mathematics. It is useless and it is a fortunetelling kind of thing. It is not science. But some countries and their useless education systems ask these "problems" to students.