Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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 x2 + y2 = 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 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) 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 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 Represent reflections, stretches, and shrinks of functions algebraically and graphically Graph in the Mirror!! 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 Over the x-axis – flips the graph of a function over the x-axis – Over the y-axis – flips the graph of a function over the y-axis – 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 5 x 9 across each axis f ( x) x2 3 Tonight’s Assignment P. 139 – 140 Ex # 3-24 m. of 3 Ex: Express h(x) so that it represents the graph of f(x) = x2 – 3 reflected over the xaxis? y-axis? Stretching & Shrinking 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). 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 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) = x2 + x - 2 Combining Transformations The order in which transformations are performed often affect the graph that results Ex 4 Use f(x) = x2 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)