Transformation of Functions

The transformation of functions includes the shifting, stretching, and reflecting of their graphs. Each of the sections below shows a different way of transforming a function. The first section focuses on vertical and horizontal shifts. The second section focuses on vertical and horizontal stretching/compressions. The final section focuses on reflecting the graph about an axis.

VERTICAL AND HORIZONTAL SHIFTS

Suppose $ c > 0 $. To obtain the graph of:

$ y = f(x) + c $: shift the graph of $ y = f(x) $ up by $ c $ units
$ y = f(x) - c $: shift the graph of $ y = f(x) $ down by $ c $ units
$ y = f(x - c) $: shift the graph of $ y = f(x) $ to the right by $ c $ units
$ y = f(x + c) $: shift the graph of $ y = f(x) $ to the left by $ c $ units

Example: Given that the equation of the black curve below is $ y = x^2 $, what are the equations for the blue and the red curves shown below?

$Graph of vertical and horizontal shifts of a quadratic function$

Solution: We are given that the equation of the black curve in the above graph is $ y = x^2 $. We notice that the blue curve is the same function, but shifted up by 10 units. Therefore, the blue function in the above graph is $ y = x^2 + 10 $. The red curve is the same function as the black one, but shifted 2 units to the right[cite: 1]. Therefore, the red function in the above graph is $ y = (x - 2)^2 $.

VERTICAL AND HORIZONTAL STRETCHES/COMPRESSIONS

Suppose $ c > 1 $. To obtain the graph of:

$ y = c f(x) $: stretch the graph of $ y = f(x) $ vertically by a factor of $ c $
$ y = \frac{1}{c} f(x) $: compress the graph of $ y = f(x) $ vertically by a factor of $ c $
$ y = f(c x) $: compress the graph of $ y = f(x) $ horizontally by a factor of $ c $[cite: 1]
$ y = f\left(\frac{x}{c}\right) $: stretch the graph of $ y = f(x) $ horizontally by a factor of $ c $

Example: Given the curve $ y = \sin(x) $ below in black, sketch the curves corresponding to $ y = 3\sin(x) $ and $ y = \sin(2x) $ and explain how each of them compares to the original function.

$Graph of original sine function$

Solution:

The black curve is the original function $ y = \sin(x) $.
The blue curve is $ y = 3\sin(x) $, stretched vertically by a factor of 3.
The red curve is $ y = \sin(2x) $, compressed horizontally by a factor of 2.

$Graph showing vertical stretch and horizontal compression of sine function$

REFLECTIONS

To obtain the graph of:

$ y = -f(x) $: reflect the graph of $ y = f(x) $ about the $ x $-axis
$ y = f(-x) $: reflect the graph of $ y = f(x) $ about the $ y $-axis

Example: Sketch the function $ y = \sqrt{x} $ and its reflections about the $ x $-axis and $ y $-axis and write the equation for each.

Solution:

$Graph of square root function and its reflections$

The black curve is the original function, $ y = \sqrt{x} $.
The blue curve shows the function reflected about the $ y $-axis, so this is the graph of $ y = \sqrt{-x} $.
The red curve shows the function reflected about the $ x $-axis, so this is the graph of $ y = -\sqrt{x} $.