Question

How do you find the function rule?

Answer 1

You need to know some base graphs:

• `f(x) =c`
• `f(x)=x`
• `f(x) = x^2`
• `f(x)=x^3`
• `f(x) = ln(x)`
• `f(x)=abs(x)`

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(x)=x^2` Pick a point on the base graph. Lets say `O(0,0)`

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(x)=(x+2)^2`), base graph will shift `-2` units on the `x`-axis. (In `x^2` `y` was `0` when `x` was `0`. But in `(x+2)^2`, `y` is `0` when `x` is `-2`)

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

`y=f(x)+2`

`y= [(x +2)^2]+2 = (x+2)^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.