1.5 Graphical Transformations Represent translations algebraically and graphically

Consider this… How is the graph (x – 2) 2 + (y+1) 2 = 16 related to the graph of x 2 + y 2 = 16?

Some change is good! Transformations - functions that map real numbers to real numbers Rigid transformations – leave the size and shape of a graph unchanged, include horizontal translations, vertical translations, reflections or any combination of these. Non-rigid transformations – generally distort the shape of a graph, include horizontal or vertical stretches and shrinks.

Vertical and Horizontal Translations Vertical translation – shift of the graph up or down in the coordinate plane Horizontal translation – shift of the graph left or right in the coordinate plane

Exploration #1 Complete the activity on p. 132 – With Partners – Use Whiteboards

Translations Let c be a positive real number. Then the following transformations result in translations of the graph of y = f(x) Horizontal translations y = f(x – c) a translation to the right by c units y = f(x + c) a translation to the left by c units Vertical translations y = f(x) + c a translation up by c units y = f(x) – c a translation down by c units

Describe how the graph of y = |x| can be transformed to the graph of the given function: a) y = |x – 4|b) y = |x| + 2

Introducing Reflections Reflections – the graphs of two functions are symmetric with respect to some line Complete Exploration #2 on p. 134 – With Partners, Use Whiteboards

Reflections Over the x-axis – Symbolically (x,y)  (x,-y) Over the y-axis – Symbolically (x,y)  (-x,y) Over the line y = x – Symbolically (x,y)  (y,x)

Reflections Transformations that result in reflections of the graph y=f(x) Across the x-axis: y = - f(x) Across the y-axis: y = f(-x)

Express h(x) so that it represents the graph of f(x) = x 2 – 3 reflected over the x-axis? y-axis?

Find an equation for the reflection of across each axis

Let’s think about this What would happen if you reflected an even function across the y-axis?

Vertical Stretches When a function is multiplied by a number whose absolute value is larger than 1, its graph is stretched vertically (Vertical Stretch by a factor of 3)(Vertical Stretch by a factor of 2)

Vertical Shrinks When a function is multiplied by a number between 0 and 1, its graph is shrunk vertically (Vertical Shrink by a factor of 1/3)(Vertical Shrink by a factor of 1/2)

Horizontal Shrinks When x is multiplied by a number greater than 1, the graph shrinks horizontally. This can look similar to a vertical stretch, but it isn’t the same thing. (Horizontal Shrink by a factor of ½ ) (Horizontal Shrink by a factor of ¼ )

Horizontal Stretches When x is multiplied by a number between 0 and 1, the graph stretches horizontally. This can look similar to a vertical shrink, but it isn’t the same thing. (Horizontal Stretch by a factor of 4) (Horizontal Stretch by a factor of 3)

Finding equations for stretches and shrinks Let y 1 = f(x) = x 3 – 16x. Find the equations for the following transformations: a) A Vertical stretch by a factor of 3 a) A Horizontal shrink by a factor of 1/2

Transform the given functions by (a) a vertical stretch by a factor of 2 and (b) horizontal shrink by a factor of 1/3. 1) f(x) = |x + 2|2) f(x) = x 2 + x - 2

Combining Transformations Use f(x) = x 2 to perform each transformation: a) A horizontal shift 2 units to the right, a vertical stretch by a factor of 3, and vertical translation 5 units up a) Apply the transformations in the reverse order

1.5 Homework Part 1: page 139 – 140 Ex # 3-24 m. of 3 Part 2: page 140 Ex #39-42, all