Shifting, scaling and reflecting are three methods of producing translations for basic graphing functions you have already learned. Knowing how to shift, scale or reflect these graphs makes you a stronger mathematics student and produces many variations on the original graphs of common functions.
The abscissa is the x-coordinate, or the distance left or right from the y-axis that allows you to locate a point using a coordinate pair. Its partner is the ordinate, or y-coordinate. This is the distance above or below the x-axis.
Using the abscissa and ordinate, you can fix a point on the coordinate graph. Then, using translations, you can move the point. Translations are performed in three ways:
In order to translate any of the common graphed functions, you need to recall and be fluent with the seven common functions themselves, presented here alphabetically because they are all equally important:
By concentrating on the original seven functions and the way they appear when graphed, you will soon develop an awareness of how each of the three translations affects the original graphed function.
Rather than burden your brain by trying to memorize the configurations of some 28 different graphs (each original and three translations), concentrate on the changes each translation provides to its original function.
The mathematics will not seem so intimidating. Rather than get lost in the details, you will come to know the graphs of all seven functions, and their translations, as a group. You will be able to do better mathematics faster, since you will save time not having to plot out individual points. Instead you will learn to recognize a given graph as, for example, the reflection of a graph of a cubic function.
Here are graphs of the seven functions. Make sure you are familiar with the shape and direction of each graph.
[insert coordinate grids showing graphs of the seven basic functions, in the same alphabetical order as the written list. The original content had six of these you can use as models: Absolute Value Function; Constant Function; Cubic Function; Greatest Integer Function or Floor Function (typical graph of this function is available at https://www.mathwarehouse.com/algebra/greatest-integer-function-and-graph.php
Linear Function or Identity Function; Quadratic Function or Squaring Function; and Square Root Function]
Each of the seven graphed functions can be translated by shifting, scaling, or reflecting:
All three functions can be combined into a single equation. Before you pick up this powerful tool, though, make certain you know all its parts:
a = the vertical stretch or compression (scale), for which |a| > 1 stretches the graph, |a| < 1 compresses the graph, and a < 0 reflects or flips the graph upside down
b = the horizontal stretch or compression (scale), for which |b| > 1 compresses the graph, |b| < 1 stretches the graph, and b < 0 reflects or flips the graph left and right
c = the horizontal shift, for which c < 0 shifts everything right and c > 0 shifts everything left
d = the vertical shift, for which d > 0 moves everything up and d < 0 moves everything down
And now, the powerful, single equation:
If you do not have a value for one of the variables, your equation still works. Do not replace the missing variable with a 0. For example, for the cubic function y = x^3, the original graph looks like this:
The translation equation, y = (-1(x^3 + 1)) + 1, which has -1 for b, 1 for c, and 1 for d but has no value for a, looks like this:
Incorrectly putting a 0 in for the a value for a produces this equation: y = 0(-1(x^3 + 1)) + 1 and looks like this:
Clearly this is an entirely different function, unrelated to your original cubic function. When in doubt, leave it out.
y = f(x) produces no translation; no values for a, b, c or d are shown.
y = f(x + 2) produces a horizontal shift to the left, because the +2 is the c value from our single equation. It is added to the x-value. For horizontal shifts, positive c values shift the graph left and negative c values shift the graph right.
y = f(x) + 2 produces a vertical translation, because the +2 is the d value. It shifts the entire graph up for positive values of d and down for negative values of d.
y = f(x - 2) + 3 gives us values for both c and d, so the translation moves 2 units right (negative c) and three units up (positive d)
y = 3 * f(x) produces a change of scale, because the absolute value of 3 (the a value in our single equation) stretches the graph.
y = -1 * f(x) = -f(x) indicates we have multiplied everything by -1; this produces a reflection across the x-axis, without changing x-axis values.
y = f(2x) shows our x-value multiplied, which means we have scaled the original function horizontally, which shrinks the graph. Horizontal changes are inverse. Multiplying by 2 actually divides every x-value by 2 to produce the y-value.
y = f(-1 * x) = f(-x) shows we multiplied only the x-value times -1; this will reflect the graph across the y-axis, without changing y-values.
Besides the issue of putting in 0s in the single equation when unnecessary, other stumbling blocks to correctly graphing translations of functions are common. Perhaps most common is keeping straight how some values are inverse, some seem backwards, and some are exactly as they appear.
To move a graph up, we add a positive value to the y-value. To move a graph down, we add a negative value to the y-value.
To move a graph right, we add a negative value to the x-value. To move a graph left, we add a positive value to the x-value.
To stretch a graph in the y-axis, we multiply the whole function times any number n such that n > 1. This will cause, for example, a vertical parabola to "open up." To compress a graph, we multiply the whole function times any number n such that 0 < n < 1. This will cause the same vertical parabola to "close up."
To stretch a graph in the x-axis, we multiply only the x-values times a constant. Larger values compress the graph and smaller values (<0 n <1) stretch the graph. This is inverse of stretching the graph in the y-axis.
To reflect a graph across the x-axis, everything is multiplied by -1.
To reflect a graph across the y-axis, only the x-value is multiplied by -1.
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