Determine a transformed equation from a graph

A transformed graph is interpreted as the image of a familiar parent function, allowing its equation to be written in the form $y=a\,f(b(x-h))+k$ , where $h$ and $k$ describe horizontal and vertical translations, $a$ describes vertical stretch, compression, or reflection, and $b$ describes horizontal scaling and reflection. The parameters are inferred from features such as vertices, intercepts, endpoints, asymptotes, and corresponding points, with attention to the reciprocal effect of horizontal scale factors; arbitrary compositions, implicit equations, and advanced function families are not included.

Detailed Explanation: Determine a transformed equation from a graph

A transformed graph can be compared with its parent function. Use the form

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

The parameters describe:

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

Remember that the horizontal scale is reciprocal: if $b=\frac12$ , the graph is stretched horizontally by a factor of $2$ .

Worked example

Suppose a graph has the shape of the exponential parent function

f(x)=2^x.

From the graph, we can identify these features:

Its horizontal asymptote is $y=-1$ .
The parent point $(0,1)$ appears at $(2,1)$ .
The parent point $(1,2)$ appears at $(4,3)$ .

Find the transformed equation.

Step 1: Determine the vertical translation

The parent function $y=2^x$ has horizontal asymptote $y=0$ .

The graph’s asymptote is $y=-1$ , so the graph has moved down $1$ unit:

k=-1.

Step 2: Determine the horizontal translation and vertical scale

The parent point $(0,1)$ has moved to $(2,1)$ .

Since the input $0$ moved to $2$ , the graph shifted right $2$ units:

h=2.

At this corresponding point, the output is still $1$ . Using

y=a\,2^{b(x-h)}+k,

substitute $(x,y)=(2,1)$ , $h=2$ , and $k=-1$ :

1=a\,2^{b(2-2)}-1.

Because $2^0=1$ ,

1=a-1,

a=2.

Step 3: Determine the horizontal scale factor

The parent point $(1,2)$ moved to $(4,3)$ .

The parent input is $1$ , so the transformed input inside the function must also be $1$ :

b(4-2)=1.

Thus,

2b=1,

and

b=\frac12.

This means the graph is stretched horizontally by a factor of $2$ .

Step 4: Write the equation

Substitute $a=2$ , $b=\frac12$ , $h=2$ , and $k=-1$ :

\boxed{y=2\cdot 2^{\frac12(x-2)}-1}.

Equivalently,

\boxed{y=2\cdot 2^{\frac{x-2}{2}}-1}.

To check, when $x=2$ ,

y=2\cdot 2^0-1=1,

and when $x=4$ ,

y=2\cdot 2^1-1=3,

which matches the two points from the graph.

Learn by doing: Determine a transformed equation from a graph

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Quadratics Vertex Form - Graph to Equation

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