1. Section 4- 3 Reflecting Graphs; Symmetry Objective: To reflect graphs and to use symmetry to sketch graphs.

2. Introduction In this section we will see the relationship between a function’s equation and its graph. –When a function’s equation is altered, its graph will predictably change We will start with the reflection of a graph. –What does the word reflection mean? mirror image –Example: Definition: Line of Reflection – located halfway between a point and its reflection Acts like a mirror Where is the line of reflection?

3. The Line of Symmetry (also called the Mirror Line) does not have to be up- down or left-right, it can be in any direction. Introduction These are the four most common lines of symmetry

4. Introduction Recall: –When a function’s equation is altered, its graph will predictably change Let’s try some examples by graphing the following. –KEY - Look for a relationship between the function’s equation and its graph. y = x 2 y = - x 2 y = x 2 - 1 Reflection over the x-axis Partial Reflection over the x-axis y = 2x - 1 y = 2(-x) - 1 Reflection over the y-axis Line of Reflection Line of Reflection Line of Reflection Line of Reflection Moral of the story – Small changes in an equation greatly change the graph

5. Introduction Recall: –When a function’s equation is altered, its graph will predictably change Let’s try some examples by graphing the following. –KEY - Look for a relationship between the function’s equation and its graph. y = x 2 y = - x 2 y = x 2 - 1 Reflection over the x-axis Partial Reflection over the x-axis y = 2x - 1 y = 2(-x) - 1 Reflection over the y-axis Line of Reflection Line of Reflection Line of Reflection Line of Reflection Moral of the story – Small changes in an equation greatly change the graph

6. Reflection in the x-axis The graph of y = -f(x) is obtained by reflecting the graph of y = f(x) in the x-axis. Notice: the point (x,y) from f(x) (the original graph) becomes the point (x,-y) on –f(x) (the reflected graph) y = f(x) y = -f(x) y = x 2 - 3 y = -(x 2 - 3) Note:The graph of is identical to the graph of y = f(x) when f(x) ≥ 0 and is identical to the graph of y = -f(x) when f(x) < 0. We will see several examples: y = f(x) y = x 2 - 3 Recall:

7. The graph y = f(-x) is obtained by reflecting the graph of y = f(x) in the y-axis. Reflection in the y-axis y = 1.5 x y = f(x) y = 1.5 -x y = f(-x) Notice: the point (x,y) from f(x) (the original graph) becomes the point (-x,y) on f(-x) (the reflected graph) y = (x + 3) 2 y = f(x) y = (-x + 3) 2 y = f(-x) Recall:

8. Reflection in the Line y = x Reflecting the graph of an equation in the line y = x is equivalent to interchanging x and y in the equation. y = x 2 Original graph and equationReflection in the line y = x x = y 2 Reflected graph and altered equation (switched x and y)

9. Reflection in the Line y = x Reflecting the graph of an equation in the line y = x is equivalent to interchanging x and y in the equation. y = x 2 Original graph and equationReflection in the line y = x x = y 2 Reflected graph and altered equation (switched x and y)

10. Symmetry A line l is called an axis of symmetry of a graph if it is possible to pair the points of a graph in such a way that l is the perpendicular bisector of the segment and the joining pair. Ex) l = axis of symmetry Recall: Let’s try a few: 1. The graph of y = f(x) is shown at the right. Sketch the graph of each of the following equations. y = f(x) a) y = -f(x) y = -f(x) y = f(x) b) y = f(x) y = f(x) c) y = f(-x) y = f(x) y = f(-x)

11. More Examples Page 135 #2 and #3

12. Homework p136-137: 1-4 (all), 6-30 (multiples of three), 31, 38 (increasing and decreasing functions) Extra Credit: 32, 33, 35