1. Graph Transformations

2. Starter Sketch each of these graphs in your book a) y = x2 b) y = x3
c) y = 1/x
d) What would y = xa look like, where a is an even number?
e) What about y = xb where b is an odd number?

3. Graph Transformationsy
y = x2 + 1
 To the right is the Graph of y = x2
 On it, we will draw the graph of
y = x2 + 1
 And now we will draw the graph of
y = x2 - 2
y = x2 x -3 -2 -1 1 2 3 y 9 4 1 1 4 9
y = x2 - 2 x
All y values increase by 1 x -3 -2 -1 1 2 3 y 10 5 2 1 2 5 10
“Adding/Subtracting a number ‘a’ from the equation itself is a vertical shift of ‘a’ units”
All y values decrease by 2 x -3 -2 -1 1 2 3 y 7 2 -1 -2 -1 2 7
f(x) represents the original graph. f(x) + 1 is the same graph, 1 unit higher

4. Graph Transformationsy
y = (x + 1)2
y = (x – 2)2
 To the right is the Graph of y = x2
 On it, we will draw the graph of
y = (x + 1)2
 And now we will draw the graph of
y = (x – 2)2 x -3 -2 -1 1 2 3 y 9 4 1 1 4 9 x
Same set of y values, for x values 1 less than before
y = x2 x -3 -2 -1 1 2 3 y 4 1 1 4 9 16
Same set of y values, for x values 2 more than before
“Adding/Subtracting a number ‘a’ before working out the y values is a horizontal shift of ‘–a’ units” x -3 -2 -1 1 2 3 y 25 16 9 4 1 1
f(x) represents the original graph f(x + 1) is the same graph, 1 unit to the left

5. Graph Transformationsy
y = 2x2
 To the right is the Graph of y = x2
 On it, we will draw the graph of
y = 2x2
 And now we will draw the graph of
y = 0.25x2
y = x2 x -3 -2 -1 1 2 3 y 9 4 1 1 4 9
y = 0.25x2 x
All y values double x -3 -2 -1 1 2 3 y 18 8 2 2 8 18
“Multiplying the equation by a number ‘a’ will stretch the graph vertically by a factor ‘a’
All y values divided by 4 x -3 -2 -1 1 2 3 y
2.25 1
0.25
0.25 1
2.25
f(x) represents the original graph. 2f(x) is the same graph, stretched vertically by a factor of 2

6. Graph Transformationsy
y = (2x)2
 To the right is the Graph of y = x2
 On it, we will draw the graph of
y = (2x)2
 And now we will draw the graph of
y = (x/3)2 x -3 -2 -1 1 2 3 y 9 4 1 1 4 9
y = (x/3)2 x
Same set of y values, for half the x values
y = x2 x -3 -2 -1 1 2 3 y 36 16 4 4 16 36
Same set of y values, for triple the x values
“Multiplying the x values by a number ‘a’ before working out the y values is a horizontal stretch by a factor 1/a” x -3 -2 -1 1 2 3 y 1
0.4
0.1
0.1
0.4 1
f(x) represents the original graph f(2x) is the same graph, but 1/2 the width

7. Translation 2 units left
Plenary
Translation 2 units left
Translation 2 units up
y = f(x) + 2
y = f(x + 2) 14 -6 -1 1

8. Summary
We have looked at how changing parts of a function will affect the graph
The best way to learn these is to talk yourself through what is happening. Think about what is being changed, and how it will affect the set of answers you get for y.