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

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

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

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

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

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

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

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

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

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

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

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

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

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?

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

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

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.

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.

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: