1. Graphical Transformations
Vertical and Horizontal Translations
Vertical and Horizontal Stretches and Shrinks

2. Take the equation f(x)= x2
How do you modify the equation to translate the graph of this equation 5 units to the right? units to the left?
How do you modify the equation to translate the graph of this equation 3 units down? units up?
What if you wanted to translate the graph of this equation 5 units to the left and 3 units down?

3. The parabola has been translated 5 units to the right.
How is the equation modified to cause this translation?

4. Notice the change in the equation y = x2 to create the horizontal
shift of 5 units to the right.
f(x) = x2
g(x) = (x-5)2

5. The parabola is now translated 5 units to the left.
How is the equation modified to cause this translation?

6. Notice the change in the graph of the equation y=x2 to
create a horizontal shift of 5 units to the left.
f(x)=x2
h(x)=(x+5)2

7. The parabola has now been translated three units down.
How is the equation modified to cause this translation?

8. Notice how the equation y = x2 has changed to make the
Vertical translation of 3 units down.
f(x)=x2
q(x)=x2-3

9. The parabola has now been translated 3 units up.
How is the equation modified to cause this translation?

10. Notice how the equation y = x2 has been changed to make the
Vertical translation 3 units up.
f(x)=x2
r(x)=x2+3

11. Write what you think would be the equation for translating
the parabola 5 units to the left and 3 units up?

12. The equation would be
What would the graph would look like?

13. g(x) is the translation of f(x) 5 units to the left and 3 units up.
f(x)= x2
g(x) = (x+5)2+3

14. Vertical and horizontal stretches and shrinks
How does the coefficient on the x2 term affect the graph of f(x) = x2?
What if we substitute an expression such as 2x into f(x)? How would that affect the graph of f(x) = x2?

15. The parabola has been vertically stretched by a factor of 2.
Notice how the equation has been modified to cause this stretch.

16. The parabola is vertically shrunk by a factor of ½.
Notice how the equation has been modified to cause this shrink.

17. By substituting an expression like 2x in for x in f(x) = x2
gives a different type of shrink. f(2x) = (2x)2. A horizontal
shrink by a factor of ½.

18. Suppose we found g(1/2x). The equation would be
y = (1/2x)2.. How would this affect the graph of the
function g(x) = x2? It is a horizontal stretch by a factor
of 2.

19. If we were to write some rules for translations of functions
and stretches/shrinks of functions, what would we write?
Horizontal translation:
Vertical translation:
Vertical stretch:
Vertical shrink:
Horizontal stretch:
Horizontal shrink: