1. 4-3 REFLECTING GRAPHS/
SYMMETRY
Objectives:
1. Reflect graphs
2. Use symmetry to sketch graphs
3. Classify even and odd functions

2. REFLECTION IN THE X-AXIS
y
= -f(x) is the reflection of y = f(x) in
the x-axis.
Original points (x, y) become (x, -y)
on the reflected graph.

3. ABSOLUTE VALUE
y
= |f(x)| is the same as y = f(x)
where f(x) 0 and same as y = -f(x)
where f(x) < 0.
Graphs never go below x axis!

4. REFLECTION IN THE Y-AXIS
y
= f(-x) is the reflection of y = f(x) in
the y-axis.
Original points (x, y) become (-x, y)
on the reflected graph.

5. REFLECTION IN THE LINE Y = X
Interchanging
x and y in an equation
results in a reflection in the line y = x.
Original points (x, y) become (y, x) on
reflected graph.

6. EXAMPLE 1
Let
f(x) = 2x - 3, sketch:
y = -f(x)
y = |f(x)|
y = f(-x)

7. EXAMPLE 2
Sketch
the graph of y = ½ x + 1 and
its reflection in y = x. Give an
equation of the reflected graph.

8. SYMMETRY
of symmetry – a line that divides
a graph such that it is a perpendicular
bisector of segments joining paired
points
Axis

9. POINT OF SYMMETRY
A
point O is a point of symmetry if it is
the midpoint of the segments joining
each pair of points on the graph.

10. QUADRATICS AND CUBICS
Parabolas
Axis
(y = ax2 + bx + c)
of symmetry at
Cubics
Point
(y = ax3 + bx2 + cx + d)
of symmetry at

11. X-AXIS SYMMETRY
(x,
-y) is on graph whenever (x, y) is.
Test:
Substitute –y for y.
If new equation = original, it has x-axis
symmetry.

12. Y-AXIS SYMMETRY
(-x,
y) is on graph whenever (x, y) is.
Test:
Substitute –x for x.
If new equation = original, it has y-axis
symmetry.

13. Y = X SYMMETRY
(y,
x) is on graph whenever (x, y) is.
Test:
Interchange x and y.
If new equation = original, it has y = x
symmetry.

14. ORIGIN SYMMETRY
(-x,
-y) is on graph whenever (x, y) is.
Test:
Substitute –x for x and –y for y.
If new equation = original, it has origin
symmetry.

15. EXAMPLE 1
Test
the equation to see which
kind(s) of symmetry it has.
x2 + xy = 4
|x|
+ |y| = 1

16. YOU TRY!
Which
have?
types of symmetry does

17. EXAMPLE 2
Use symmetry
of y4 = x + 1
to sketch the graph

18. EXAMPLE 3
Find
the point of symmetry of the
graph of f(x) = -x3 + 15x2 – 48x + 45
The
function has a local min at (2, 1).
Where does a local max occur?