Skill: Graph transformed functions from a parent function

A transformed function is easiest to graph by comparing it with its parent function. In

$$y=a f\bigl(b(x-h)\bigr)+k,$$

• $h$ shifts the graph horizontally: right if $h>0$, left if $h<0$.
• $k$ shifts the graph vertically: up if $k>0$, down if $k<0$.
• $a$ affects vertical stretch or compression. If $a<0$, it also reflects the graph across the $x$-axis.
• $b$ affects horizontal scaling. A larger $\vert b \vert$ compresses the graph horizontally, and a negative $b$ reflects it across the $y$-axis.

The safest way to handle all the changes is to map points from the parent graph:

$$(x,y)\longmapsto \left(h+\frac{x}{b},\ ay+k\right).$$

The division by $b$ explains why horizontal changes work opposite to what appears inside the function.

Example

Graph

$$y=-2f\bigl(2(x-1)\bigr)+3, \qquad \text{where } f(x)=x^2.$$

1. Identify the parent function

Since $f(x)=x^2$, the parent graph is

$$y=x^2.$$

It has a vertex at $(0,0)$ and opens upward.

2. Identify the transformations

Compare the equation with

$$y=a f\bigl(b(x-h)\bigr)+k.$$

Here,

$$a=-2,\qquad b=2,\qquad h=1,\qquad k=3.$$

Therefore:

• $h=1$: shift right $1$ unit.
• $k=3$: shift up $3$ units.
• $a=-2$: reflect across the $x$-axis and stretch vertically by a factor of $2$.
• $b=2$: compress horizontally by a factor of $2$.

3. Choose points on the parent graph

Use several familiar points from $y=x^2$:

$$(-2,4),\quad (-1,1),\quad (0,0),\quad (1,1),\quad (2,4).$$

4. Apply the mapping

Use

$$(x,y)\longmapsto \left(1+\frac{x}{2},\ -2y+3\right).$$

For example, the parent vertex $(0,0)$ maps to

$$\left(1+\frac{0}{2},-2(0)+3\right)=(1,3).$$

Map all the points:

$$\begin{array}{c|c} \text{Parent point} & \text{Transformed point}\\ \hline (-2,4) & (0,-5)\\ (-1,1) & \left(\frac12,1\right)\\ (0,0) & (1,3)\\ (1,1) & \left(\frac32,1\right)\\ (2,4) & (2,-5) \end{array}$$

5. Draw the graph

Plot the transformed points:

$$(0,-5),\quad \left(\frac12,1\right),\quad (1,3),\quad \left(\frac32,1\right),\quad (2,-5).$$

Connect them with a smooth parabola. The vertex is $(1,3)$, the parabola opens downward because $a$ is negative, and it is narrower than the parent graph because of the vertical stretch by $2$ and horizontal compression from $b=2$.

Thus, the graph is a downward-opening parabola with vertex

$$\boxed{(1,3)}.$$