Topics:
• Transformations of
Graphs of Functions
• Symmetric about the yaxis, x-axis or origin
• Is a function odd, even or
neither?
MAT 150
Algebra
Class #12
Parent Functions
Function
Identify
𝑦=𝑥
Linear
𝑦 = 𝑥2
Quadratic
𝑦=
𝑥
Square Root
1
𝑦=
𝑥
𝑦 = 2𝑥
Rational
𝑦 = log10 𝑥
Logarithm
𝑦= 𝑥
Absolute Value
𝑦 = 𝑥3
Cubic
Power
Sketch
Transformation of Graphs of
Functions
Type of Shift
Mathematical
Language
Verbal
Language
Vertical Shifts
y = f(x) + k
Graph is shifted k units
up if k > 0 and k units
down if k< 0
Horizontal Shifts
y = f(x – h)
Graph is shifted h units
right if h > 0 and h units
left if h < 0.
Stretching
y = a* f(x)
Graph is vertically
stretched using a
factor of |a| if |a|> 1.
Transformation of Graphs of
Functions
Type of Shift
Mathematical
Language
Verbal
Language
Compressing
y = a* f(x)
Graph is vertically
stretched using a factor
of |a| if |a|< 1.
Reflections x-axis
y = -f(x)
Graph is reflected
across the x-axis.
Reflection y-axis
Y = f(-x)
Graph is reflected
across the y-axis.
Example
1
Use the function 𝑦 = −
𝑥 + 4 − 10 to
2
answer the following questions.
a.
b.
c.
The graph of this function is a shift graph
of which basic function?
Describe the transformations used to
obtain the graph of the function.
Sketch a graph of the function.
Symmetry
Symmetry
Mathematical
Language
Verbal
Language
Respect to y-axis
f(-x) = f(x)
If every point (x, y) on the graph,
the point (-x , y) is also on the
graph. Such a function is called
an EVEN FUNCTION.
Respect to the origin
f(-x) = -f(x)
If every point (x, y) on the graph,
the point (-x , -y) is also on the
graph. Such a function is called
an ODD FUNCTION.
Respect to x-axis
If every point (x, y) on the graph,
the point (x , - y) is also on the
graph.
Examples
Determine algebraically whether the graph
of the given equation is symmetric with
respect to x-axis, y-axis and/or the origin. Is
the function even, odd or neither? Confirm
graphically.
6
𝑥
a.
𝑓(𝑥) =
b.
𝑔(𝑥) = 𝑥 2 − 2
𝑤 𝑥 = 2𝑥 − 𝑥 2
c.
Assignment
Pg. 259
#3-9odd
#17-18
#23 & 25
#27-35 odd
#51, 60