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 Linearf(x) = mx + b Quadraticf(x) = x 2 Square Rootf(x) = √x Exponentialf(x) = b x Rationalf(x) = 1/x Logarithmicf(x) = log b x Absolute Valuef(x) = |x|

4. Types of Transformations

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)+k is a transformation of 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 RIGHT of 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)+k is a translation of k units UP of the graph of y = f (x). Example: f(x) = x 2 + 3 If k<0, then the graph of y=f(x)+k is a translation of |k| units DOWN of the graph of y = f ( x). Example: f(x) = x 2 - 4

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

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 Horizontal Translation Horizontal S t r e t c h or compression

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 InversesAsymptotes

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 y values to find the coordinates of an inverse function. When given an equation, interchange the x and y variables, 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.