Determine transformed equations from graphs

From a graph and a known parent function, the learner infers an equation such as $y=a\,f(b(x-h))+k$ , identifying vertical and horizontal translations, reflections, and stretches or compressions from corresponding features such as vertices, intercepts, endpoints, or asymptotes. The reasoning accounts for the reversed effect of horizontal changes inside the function and distinguishes transformations of a graph from changes to the function’s rule; arbitrary nonlinear changes of variables and abstract transformation theory are outside this scope.

Detailed Explanation: 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 $a$ .

Learn by doing: Determine transformed equations from graphs

Click a topic below to practice the foundational skills you'll need, learn the steps, or master this skill

Practice with unlimited practice problems

Function Transformations (Definition) - Single Transformation Graph to Function

Earned ?