1. Reflections and Symmetry Section 4.3

2. Reflection in the x-axis

3. y=–f(x) (x,y) maps to (x, –y)

4. Graph y = |f(x)| When f(x) ≥ 0, the graph is identical to y = f(x) When f(x) < 0, the graph is identical to y = –f(x) The dependent variable will NEVER be negative.

5. Reflection in the y-axis

6. y=f(–x) (x,y) is mapped to (–x,y)

7. Reflection in the line y=x (x,y) is mapped to (y, x)

8. Symmetry A line is called an axis of symmetry if it is possible to pair points of the graph in such a way that the line is the perpendicular bisector of the segment A point, O, is called a point of symmetry if it pairs points of the graph in such a way that the point, O, is the midpoint of the segment joining the pair. 180° point rotation

9. To find line/point symmetr y To find the axis of symmetry in a quadratic function x = –b/(2a) Find the line symmetry in f(x) = 3x 2 -2x+8 x=–b/(2a) = –(–2)/(2*3) = 2/6 X = 1/3 To find the point of symmetry in a cubic function x = –b/(3a) Find the point of symmetry of f(x) = -x 3 +15x 2 -48x+45 x = –b/3a = -(15)/(–3*1) = 5 F(5) =(– 5) 3 +15(5)) 2 –48(5)+45= 55 the point is (5, 55) If (2,1) falls on the graph, find another point that also lies on the graph If (5,55) is the midpoint of the segment containing (2,1) and (x,y) then the point must be (8, 109) WHY?

10. Symmetry Tests for functions Y – axis symmetry (x, y) pairs to (–x, y) X – axis symmetry (x,y) pairs to (x, –y) Y=x line symmetry (x, y) pairs to (y, x) 180° point rotation around (0, 0) (x, y) pairs to (–x, –y) Check symmetry by substituting in the mapping. If the expression simplifies to the original expression then the graph has that type of symmetry.

11. Determine the symmetry in x=y 4 X axis – substitute (x, –y) X=(–y) 4 Y axis – substitute (–x, y) –x = y 4 Y=x – substitute (y, x) X=y 4 180° Point (0.0) - substitute (–x, –y) –x = (–y) 4

12. Graph of x=y^4