1 5 Graphical Transformations l Represent translations algebraically

1. 5 Graphical Transformations l Represent translations algebraically and graphically

Consider this… l 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!! l l l 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 l l 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 l l l Complete the activity on p. 132 No talking – first 4 min. You will be able to discuss with classmates the last 2 min.

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

Ex 1 Describe the graph of y = |x| can be transformed to the graph of the given function: a) y = |x – 4| b) y = |x| + 2

Reflections, Stretches, and Shrinks l Represent reflections, stretches, and shrinks of functions algebraically and graphically

Graph in the Mirror!! l l Reflections – the graphs of two functions are symmetric with respect to some line Complete Exploration #2 on p. 134 First 6 min (No Talking) Last 2 min (Discuss with a neighbor)

Reflections l Over the x-axis – flips the graph of a function over the x-axis – l Over the y-axis – flips the graph of a function over the y-axis – l Symbolically (x, y) (x, -y) Symbolically (x, y) (-x, y) Over the line y = x – flips the graph of a function over the line y = x – Symbolically (x, y) (y, x)

Ex 1 Find an equation for the reflection of across each axis

Tonight’s Assignment l P. 139 – 140 Ex # 3 -24 m. of 3

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

Stretching & Shrinking l l l Complete the exploration on p. 136 First 8 min. No talking Last 8 min. you can discuss with a neighbor

Stretches and Shrinks Let c be a positive real number. The following transformations result in stretches or shrinks of the graph of y = f(x). l Horizontal stretches or shrinks y = f(x/c) a stretch by a factor of c if c > 1 a shrink by a factor of c if c < 1 l Vertical stretches or shrinks y = cf(x) a stretch by a factor of c if c > 1 a shrink by a factor of c if c < 1

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

Combining Transformations l The order in which transformations are performed often affect the graph that results Ex 4 Use f(x) = x 2 to perform each transformation. Write the formula for the resulting function. a) A horizontal shift 2 units to the right, a vertical stretch by a factor of 3, and vertical translation 5 units up Apply the transformations in the reverse order Are the graphs the same? Are the formulas the same? b) c)