1. Precalculus Lesson 1.4 Shifting, Reflecting, and Stretching Graphs

2. An Important Question How are the graphs of new functions related to the graph of the original function? You will need to ask yourself: Is the new graph vertically shifted up or down? Is the new graph horizontally shifted left or right? Is the new graph vertically stretched by pulling the graph away from the x-axis? Is the new graph vertically compressed by pushing the graph towards the x-axis? Is the new graph horizontally stretched by pulling the graph away from the y-axis? Is the new graph horizontally compressed by pushing the graph towards the y-axis?

3. f(x) = x 3 +3x 2 We are going to look at what happens when the new function is: g(x) = f(x) + 2 h(x) = f(x) – 5 j(x) = f(x-4) k(x) = f(x+5) m(x) = 4f(x) n(x) = -3f(x) p(x) = f(2x) r(x) = f(-x) s(x) = 0.5f(x) t(x) = -.25f(x) u(x) = f(0.5x) v(x) = f(-0.25x)

4. Case #1 f(x) = x 3 +3x 2 g(x) = f(x) + 2 How is the graph of g(x) related to the graph of f(x)? xY 1 = f(x)Y 2 = g(x) -2 0 1 2 3 4

5. Case #2 f(x) = x 3 +3x 2 h(x) = f(x) – 5 How is the graph of h(x) related to the graph of f(x)? xY 1 = f(x)Y 2 = h(x) -2 0 1 2 3 4

6. Case #3 f(x) = x 3 +3x 2 j(x) = f(x-4) How is the graph of j(x) related to the graph of f(x)? xY 1 = f(x)Y 2 = j(x) -3 -2 0 1 2 3 4

7. Case #4 f(x) = x 3 +3x 2 k(x) = f(x+5) How is the graph of k(x) related to the graph of f(x)? xY 1 = f(x)Y 2 = k(x) -7 -6 -5 -4 -3 -2 0

8. Case #5 f(x) = x 3 +3x 2 m(x) = 4f(x) How is the graph of m(x) related to the graph of f(x)? xY 1 = f(x)Y 2 = m(x) -4 -3 -2 0 1 2

9. Case #6 f(x) = x 3 +3x 2 n(x) = -3f(x) How is the graph of n(x) related to the graph of f(x)? xY 1 = f(x)Y 2 = n(x) -4 -3 -2 0 1 2

10. Case #7 f(x) = x 3 +3x 2 p(x) = f(2x) How is the graph of p(x) related to the graph of f(x)? xY 1 = f(x)Y 2 = p(x) -4 -3 -2 0 1 2

11. Case #8 f(x) = x 3 +3x 2 r(x) = f(-x) How is the graph of r(x) related to the graph of f(x)? xY 1 = f(x)Y 2 = r(x) -3 -2 0 1 2 3

12. Case #9 f(x) = x 3 +3x 2 s(x) = 0.5f(x) How is the graph of r(x) related to the graph of f(x)? xY 1 = f(x)Y 2 = s(x) -4 -3 -2 0 1 2

13. Case #10 f(x) = x 3 +3x 2 t(x) = -.25f(x) How is the graph of r(x) related to the graph of f(x)? xY 1 = f(x)Y 2 = t(x) -4 -3 -2 0 1 2

14. Case #11 f(x) = x 3 +3x 2 u(x) = f(0.5x) How is the graph of r(x) related to the graph of f(x)? xY 1 = f(x)Y 2 = u(x) -5 -4 -3 -2 0 1

15. Case #12 f(x) = x 3 +3x 2 v(x) = f(-0.25x) How is the graph of r(x) related to the graph of f(x)? xY 1 = f(x)Y 2 = v(x) -2 0 1 2 3 4 5 6 7 8

16. Generalized Questions What happens to the graph of a function when: You add to the output of the function? You subtract from the output of the function? You add to the input of the function? You subtract from the input of the function? You multiply the output of the function by a positive number greater than or equal to 1? You multiply the output of the function by a positive number less than 1? You multiply the output of the function by a negative number less than or equal to -1? You multiply the output of the function by a negative number greater than -1? You multiply the input of the function by a positive number greater than or equal to 1? You multiply the input of the function by a positive number less than 1? You multiply the input of the function by a negative number less than or equal to -1? You multiply the input of the function by a negative number greater than -1?

17. New FunctionEffect on the Graph of the Original Function Y = -f(x) Y = f(-x) Y = f(x) – c (c>0) Y = f(x) + c (c>0) Y = f(x-c) (c>0) Y = f(x+c) (c>0) Y = c*f(x) (0<c<1) Y = c*f(x) (c>1) Y = f(cx) (0<c<1) Y = f(cx) (c>1)