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Graph transformed functions from a parent function

A transformed function is interpreted in relation to its parent graph: in y=af(b(x−h))+ky=a f(b(x-h))+k, hh and kk produce horizontal and vertical translations, aa controls vertical stretch or compression and reflection, and bb controls horizontal scaling and possible reflection. The corresponding mapping of key points clarifies why horizontal changes act oppositely inside the input and supports graphing familiar parent functions such as linear, quadratic, absolute-value, square-root, and reciprocal functions; abstract transformations of function spaces and advanced parameter analysis are not included.

Detailed Explanation: Graph transformed functions from a parent function

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

y=af(b(x−h))+k,y=a f\bigl(b(x-h)\bigr)+k,
  • hh shifts the graph horizontally: right if h>0h>0, left if h<0h<0.
  • kk shifts the graph vertically: up if k>0k>0, down if k<0k<0.
  • aa affects vertical stretch or compression. If a<0a<0, it also reflects the graph across the xx-axis.
  • bb affects horizontal scaling. A larger ∣b∣\vert b \vert compresses the graph horizontally, and a negative bb reflects it across the yy-axis.

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

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

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

Example

Graph

y=−2f(2(x−1))+3,where f(x)=x2.y=-2f\bigl(2(x-1)\bigr)+3, \qquad \text{where } f(x)=x^2.

1. Identify the parent function

Since f(x)=x2f(x)=x^2, the parent graph is

y=x2.y=x^2.

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

2. Identify the transformations

Compare the equation with

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

Here,

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

Therefore:

  • h=1h=1: shift right 11 unit.
  • k=3k=3: shift up 33 units.
  • a=−2a=-2: reflect across the xx-axis and stretch vertically by a factor of 22.
  • b=2b=2: compress horizontally by a factor of 22.

3. Choose points on the parent graph

Use several familiar points from y=x2y=x^2:

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

4. Apply the mapping

Use

(x,y)⟼(1+x2, −2y+3).(x,y)\longmapsto \left(1+\frac{x}{2},\ -2y+3\right).

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

(1+02,−2(0)+3)=(1,3).\left(1+\frac{0}{2},-2(0)+3\right)=(1,3).

Map all the points:

Parent pointTransformed point(−2,4)(0,−5)(−1,1)(12,1)(0,0)(1,3)(1,1)(32,1)(2,4)(2,−5)\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),(12,1),(1,3),(32,1),(2,−5).(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)(1,3), the parabola opens downward because aa is negative, and it is narrower than the parent graph because of the vertical stretch by 22 and horizontal compression from b=2b=2.

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

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

Learn by doing: Graph transformed functions from a parent function

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