Transformations of the Parent Functions

1. Transformations of the Parent Functions

2. What is a Parent Function • A parent function is the most basic version of an algebraic function.

3. Types of Parent Functions • Linear f(x) = mx + b • Quadratic f(x) = x2 • Square Root f(x) = √x • Exponential f(x) = bx • Rational f(x) = 1/x • Logarithmic f(x) = logbx • Absolute Value f(x) = |x|

4. Types of Transformations • Vertical Translations • Vertical S t r e t c h • Vertical Compression • Reflections • Over the x-axis

5. ….More Transformations • Horizontal Translations • Horizontal S t r e t c h • Horizontal Compression • Reflections • Over the y-axis

6. FAMILIES TRAVEL TOGETHER…… • Families of Functions • If a, h, and k are real numbers with a≠0, then the graph of y = a f(x–h)+kis a transformationof the graph of y = f ( x). • All of the transformations of a function form a family of functions. • F(x) = (a - h)+ k – Transformations should be applied from the “inside – out” order.

7. Horizontal Translations • If h > 0, then the graph of y = f (x – h) is a translation of h units to the RIGHTof the graph of the parent function. • Example: f(x) = ( x – 3) • If h<0,then the graph of y=f(x–h) is a translation of |h|units to the LEFT of the graph of parent function. • Example: f(x) = (x + 4) • *Remember the actual transformation is (x-h), and subtracting a negative is the same as addition.

8. Vertical Translations • If k>0, then the graph of y=f(x)+kis a translation of kunits UPof the graph of y = f (x). • Example: f(x) = x2 +3 • If k<0, then the graph of y=f(x)+kis a translation of|k| units DOWNof the graph of y = f ( x). • Example: f(x) = x2- 4

9. Vertical Stretch or Compression • The graph of y = af( x) is obtained from the graph of the parent function by: • stretchingthe graph of y = f ( x) by awhen a > 1. • Example: f(x) = 3x2 • compressing the graph of y=f(x) by a when 0<a<1. • Example: f(x) = 1/2x2

10. Reflections • The graph of y = -a f(x) is reflected over the y-axis. • The graph of y = f(-x) is reflected over the x-axis.

11. Transformations - Summarized Y = a f( x-h) + k Vertical S t r e t c h or compression Horizontal S t r e t c h or compression Horizontal Translation Vertical Translation

12. Multiple Transformations • Graph a function involving more than one transformation in the following order: • Horizontal translation • Stretching or compressing • Reflecting • Vertical translation

13. Are we there yet? • Parent Functions • Function Families • Transformations • Multiple Transformations • Inverses • Asymptotes

14. Where do we go from here? • Inverses of functions • Inverse functions are reflected over the y = x line. • When given a table of values, interchange the x and yvalues to find the coordinates of an inverse function. • When given an equation, interchange the x and yvariables, and solve for y.

15. Asymptotes • Boundary line that a graph will not cross. • Vertical Asymptotes • Horizontal Asymptotes • Asymptotes adjust with the transformations of the parent functions.