4.3 Reflecting Graphs; Symmetry4.3 Reflecting Graphs; SymmetryObjective To reflect graphs and use symmetry to sketch graphs.

Be able to test equations for symmetry.

Use equations to describe reflections and translations of graphs.

Line of Reflection◦ acts like a mirror◦ located halfway between a point and its reflection

Reflecting Across:1.x-axis

2.y-axis

3.line y = x

**NOTE: points on the line itself do not move when reflected.

ReflectionsReflections

X-Axis ReflectionX-Axis ReflectionThe x-axis acts like a mirror. Only the The x-axis acts like a mirror. Only the ‘y’ value changes into the opposite.‘y’ value changes into the opposite.

Notice how (–1, 3) became (–1, –3) Notice how (0, 6.5) became (0, –6.5)

When (x, y) is on the original, (x, -When (x, y) is on the original, (x, -y) becomes the point on the y) becomes the point on the reflected graph.reflected graph.

1. The graph of can be obtained by reflecting the graph of in the x-axis. Algebraically, to obtain a reflecting graph of y = f(x), we only need to multiply on the original function.

Reflections in the Reflections in the xx-axis-axisy = –f(x)

y = f(x)

by negative 1

2. The graph of is keeping the graph of y = f(x) when and reflecting the graph of when The graph of has no dip below the x-axis. So graph of only flips the negative portion of graph of

y = | f(x)|

f(x) ≥ 0 y = f(x) f(x) < 0.

y = | f(x)|y = | f(x)|

y = f(x).

y = -f(x)y = -f(x)

Y-Axis ReflectionY-Axis ReflectionThe y-axis acts like a mirror. Only the The y-axis acts like a mirror. Only the ‘x’ value changes into the opposite.‘x’ value changes into the opposite.

Notice how (-5, -6) became (5, -6)

When (x, y) is on the original, (-x, When (x, y) is on the original, (-x, y) becomes the point on the y) becomes the point on the reflected graph.reflected graph.

Reflection in the Reflection in the yy-axis-axis3. The reflection graph of about the

y-axis can be obtained algebraically by the graph of

y = f(x)

y = f(-x)

Given the graph is y = f(x).Given the graph is y = f(x).Sketch y = f(-x)Sketch y = f(-x)

y = f(-x)y = f(-x)

Reflection in the line y = xReflection in the line y = xReflecting the graph of an equation in the line y = Reflecting the graph of an equation in the line y = x is equivalent to interchanging x and y in the x is equivalent to interchanging x and y in the

equation.equation.

When (x, y) is on the original, (y, When (x, y) is on the original, (y, x) becomes the point on the x) becomes the point on the reflected graph.reflected graph.

Notice how (2, -4) became (-4, 2)

We also know this as the inverse We also know this as the inverse f f -1-1(x).(x).

Given the graph is y = f(x).Given the graph is y = f(x).Sketch f Sketch f -1-1(x)(x)

y = f(-x)y = f(-x)

Given the following functions. Given the following functions. Write the equation after it is Write the equation after it is reflected over the x-axis, y-reflected over the x-axis, y-axis, and the line y = x.axis, and the line y = x.1)1)y = 6x + 5y = 6x + 5

x-axisx-axis y-axisy-axisy = xy = x

2) y = (x – 4)2 – 2

x-axisx-axis y-axisy-axisy = xy = x

x-axisx-axis y-axisy-axisy = xy = x

HOMEWORK:HOMEWORK:Textbook p.136, #1-4, 7–Textbook p.136, #1-4, 7–12 12 (Write the equations of (Write the equations of reflection over x-axis, y-reflection over x-axis, y-axis and y = x line.)axis and y = x line.)