5 Steps To Graph Function Transformations In Algebra

1. Identify The Parent Function

• Everything you?re expected to graph on your own is based on a more basic graph (parent function) that you NEED to memorize
• Look at the image above and check with your teacher to see which you are responsible for

2. Reflect Over X-Axis or Y-Axis

• If there is a negative outside parentheses, then reflect over the x-axis, or vertically (all the y-values become negative)
• ex: f(x) = -x2
• If there is a negative inside parentheses, then reflect over the y-axis horizontally (all the x-values become negative) and factor out the negative
• ex: f(x) = 1 / (-x+3) becomes f(x) = 1 / (-(x-3)) **THIS IS TRICKY**

3. Shift (Translate) Vertically or Horizontally

• If there is a number being added outside parentheses, then shift it up vertically by that amount, or if the number is being subtracted, then shift it down
• ex: f(x) = x2?2 goes down 2
• If there is a number being added inside parentheses, then shift it left by that amount, or if the number is being subtracted, then shift it right (opposite what you?d expect)
• ex: f(x) = (x-2)2 is shifted to the right 2 units

4. Vertical and Horizontal Stretches/Compressions

• If there is a whole number coefficient outside parentheses, then multiply the y-values of all points by that coefficient and see the graph stretch vertically
• ex: f(x) = 4×2 is stretched vertically by a factor of 4
• If there is a fractional coefficient outside parentheses, then multiply the y-values of all points by that coefficient and see the graph compress vertically
• ex: f(x) = 1/2 x2 is compressed vertically by a factor of 2
• If there is a whole number coefficient inside parentheses, then multiply the x-values of all points by the inverse of (!) that coefficient and see the graph stretch horizontally (opposite of what you?d expect)
• ex: f(x) = (4x)2 is compressed horizontally by a factor of 4
• If there is a fractional coefficient inside parentheses, then multiply the x-values of all points by the inverse of (!) that coefficient and see the graph compress horizontally (opposite of what you?d expect)
• ex: f(x) = (1/2 x ? 4)2 becomes (1/2 (x ? 8))2 is stretched horizontally by a factor of 2

5. Plug in a couple of your coordinates into the parent function to double check your work

• REMEMBER: A GRAPH IS JUST A SET OF POINTS THAT SATISFY AN EQUATION
• That means you can always check your work by plugging in an x-value (I recommend x=0, and seeing if the y-value fits the y-value of your graph)

QUICK REVIEW:

1. Reflect > Shift > Stretch
2. Inside parentheses = think opposite for stretches and shifts and FACTOR (if necessary)
3. Inside parentheses = horizontal changes (flip over y-axis)
4. Outside parentheses = vertical changes (flip over x-axis)