Skill: Determine transformed equations from graphs

To determine an equation from a graph, compare the graph with its known parent function. Use the form

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

The parameters describe the transformations:

• $h$: horizontal translation
• $k$: vertical translation
• $a$: vertical stretch, compression, or reflection
• $b$: horizontal stretch, compression, or reflection

Remember that changes inside the function have the opposite effect: $b=\frac12$ creates a horizontal stretch by a factor of $2$.

Example

Suppose the parent function is

$$f(x)=x^2,$$

and the graph is an upward-opening parabola with:

• vertex $(3,-1)$
• points $(1,1)$ and $(5,1)$

We will determine its equation.

Step 1: Identify the translations

The parent parabola $y=x^2$ has vertex $(0,0)$. The new vertex is $(3,-1)$, so the graph moves:

• $3$ units right, giving $h=3$
• $1$ unit down, giving $k=-1$

The equation now has the form

$$y=a\,f\bigl(b(x-3)\bigr)-1.$$

Step 2: Determine the horizontal change

On the parent graph, the points one unit from the vertex are

$$(-1,1)\quad\text{and}\quad(1,1).$$

On the new graph, the corresponding points are $(1,1)$ and $(5,1)$. From the new vertex’s $x$-coordinate, $3$, these points are $2$ units away:

$$5-3=2.$$

The horizontal distance has doubled, so the graph has a horizontal stretch by a factor of $2$. Therefore,

$$b=\frac12.$$

This gives

$$y=a\,f\left(\frac12(x-3)\right)-1.$$

Step 3: Determine the vertical change

Use the point $(5,1)$. Since

$$f(1)=1,$$

the input to $f$ is correct because

$$\frac12(5-3)=1.$$

Substitute the point into the equation:

$$1=a(1)-1.$$

Thus,

$$a=2.$$

Step 4: Write the equation

The transformed equation is

$$\boxed{y=2f\left(\frac12(x-3)\right)-1}.$$

Since $f(x)=x^2$, this can also be written as

$$\boxed{y=2\left(\frac12(x-3)\right)^2-1}.$$

To check, when $x=5$,

$$y=2\left(\frac12(5-3)\right)^2-1 =2(1)^2-1 =1,$$

which matches the graph.

When reading a graph, first use a key feature such as a vertex or endpoint to find $h$ and $k$. Then compare corresponding points to determine the horizontal factor $b$ and vertical factor $a$.