Chapter 3 The Nature of Graphs
3.1 Symmetry and Coordinate Graphs
Symmetry with Respect to the Origin
Ex 1: Determine whether each graph is symmetric with respect to the origin.
𝑥
a. 𝑓(𝑥) = 𝑥 5
b. 𝑔(𝑥) =
1−𝑥
Symmetry with respect to the x-axis
Symmetry with respect to the y-axis
Symmetry with respect to 𝒚 = 𝒙
Symmetry with respect to 𝒚 = −𝒙
Ex 2: Determine whether the graph of 𝑥𝑦 = −2 is symmetric with respect to the x-axis, y-axis, the line 𝑦 = 𝑥,
the line 𝑦 = −𝑥, or none of these
Ex 3: Determine whether the graph of 𝑥 2 + 𝑦 = 3 is symmetric with
respect to the x-axis, y-axis, both, or neither. Use the information
about the equation’s symmetry to graph the relation
Even Functions
Odd Functions
3.2 Families of Graphs
Each function has a corresponding “__________________________________”. The parent function is one
before any transformation has occurred.
A ________________________________ move the graph in the four cardinal directions.
A_______________________________ creates a mirror image across a line.
A _______________________________ stretches or shrinks a graph vertically or horizontally.
Each parent function reacts to the transformations the same way. That means that given some parent
function, 𝑓(𝑥), then 𝑓(𝑥) + 1 affects each graph the same way…it doesn't matter if the function is quadratic,
a square root, or absolute value.
Ex 1: Graph 𝑓(𝑥) = |𝑥| and 𝑔(𝑥) = −|𝑥|. Describe how the graphs of 𝑓(𝑥) and 𝑔(𝑥) are related.
Given 𝑓(𝑥) here is how we apply each transformation
Transformation
Translation – Vertical
Translation – Horizontal
Reflection – x-axis
Reflection – y-axis
Dilation – Vertical
Dilation – Horizontal
Note:
Function Notation
Layman’s Terms
Ex 2: Describe each function as a transformation of the graph 𝑦 = 𝑥 2 . Then sketch the graph of the each
function.
a. 𝑦 = 𝑥 2 + 1
b. 𝑦 = (𝑥 − 2)2
c. 𝑦 = −2(𝑥 + 1)2 + 4
Ex 3: Describe how each pair of functions is related.
2
a. 𝑦 = |𝑥| and 𝑦 = |𝑥|
3
b. 𝑦 = |𝑥| and 𝑦 = 2.5|𝑥|
c. 𝑦 = √𝑥 and 𝑦 = √𝑥 + 1 + 4
d. 𝑦 = 𝑥 3 and 𝑦 = −2(𝑥 + 3)3
3.3 Graphs of Nonlinear Inequalities
Ex 1: Graph 𝑦 ≥ (𝑥 − 4)2 − 2
Ex 2: Graph 𝑦 > 3 − |𝑥 + 2|
Ex 3: Solve |𝑥 − 2| − 5 < 4
3.4 Inverse Functions and Relations
Inverse Function
The inverse of a function is the transpose of all points __________________________ of the original function
How do we tell if a graph is a function?
How can we tell if the inverse of a graph is a function?
Ex 1: Consider 𝑓(𝑥) = (𝑥 − 2)2 + 4
a. Graph 𝑓(𝑥)
b. Is 𝑓 −1 (𝑥) a function?
c. Find 𝑓 −1 (𝑥)
d. Graph 𝑓 −1 (𝑥)
Ex 2: Graph each function and its inverse
a. 𝑦 = ±√𝑥 + 3 − 1
b. 𝑦 = 2|𝑥 + 3| − 1
Determining if two functions are inverse functions
Two functions, 𝑓 and 𝑓 −1 are inverses if and only if
3
𝑥+4
Ex 3: Given 𝑓(𝑥) = 2𝑥 3 − 4 and 𝑔(𝑥) = √
2
determine if they are inverses. Explain how you know.
3.5 Continuity and End Behavior
Types of Discontinuity
Continuity Test
A function is continuous at 𝑥 = 𝑐 if it satisfies the following conditions:
1.
2.
3.
Ex 1: Determine whether each function is continuous at the given x-value.
a. 𝑓(𝑥) = 3𝑥 2 + 7 at 𝑥 = 1
𝑥−2
b. 𝑓(𝑥) = 𝑥 2 −4 at 𝑥 = 2
1
𝑖𝑓 𝑥 > 1
c. 𝑓(𝑥) = {𝑥
at 𝑥 = 1
𝑥 𝑖𝑓 𝑥 ≤ 1
End Behavior
The end behavior of a function describes what the _______________ do as ________ becomes greater and
greater
End Behavior of Polynomial Functions
An: positive, n: even
An: negative, n: even
An: positive, n: odd
An: negative, n: odd
3.6 Critical Points and Extrema
Definitions
 Critical points – points on a graph at which a line drawn tangent to the curve is ___________________
or _____________________
 Maximum – a critical point. When the graph of a function is ________________________ to the left of
x = c and _________________________to the right of x = c , then there is a maximum at x = c
o Two types:
o Ex:

Minimum – a critical point. When the graph of a function is ________________________ to the left of
x = c and _________________________to the right of x = c , then there is a minimum at x = c
o Two Types:
o Ex:

Point of Inflection – a critical point. When the graph changes its curvature.
o Ex:

Extremum – the general term for ______________________ or ________________________
o Two Types:
Ex 1: Locate the extrema for the graph of 𝑦 = 𝑓(𝑥). Name and classify
the extrema of the function.
Ex 2: Locate the extrema for the graph of 𝑦 = 𝑓(𝑥). Name and classify
the extrema of the function.
Derivative of a polynomial
Ex 3: Find the first and second derivatives of the following polynomials
1
a. 𝑓(𝑥) = 5𝑥 4 + 3𝑥 3 − 2𝑥
b. 𝑔(𝑥) = −3𝑥 6 + 2 𝑥 4 − 5𝑥 2 + 2
Ex 4: Using derivative and end behavior, graph 𝑓(𝑥) = 5𝑥 3 − 10𝑥 2 — 20𝑥 + 7 and determine and classify its
extrema.
Ex 5: Using derivative and end behavior, graph 𝑓(𝑥) = 𝑥 3 = 8𝑥 + 3 and determine and classify its extrema.
Ex 6: The function 𝑓(𝑥) = 2𝑥 5 − 5𝑥 4 − 10𝑥 3 has critical points at 𝑥 = −1, 𝑥 = 0, 𝑥 = 3. Determine whether
each of these critical points is the location of a maximum, a minimum, or a point of inflection.
3.7 Graphs of Rational Functions
Asymptotes
A line that the graph approaches but never intersects
3 types:
1.
2.
3.
Ex 1: Determine the asymptotes for the graph of 𝑓(𝑥) =
Ex 2: Determine the slat asymptote for 𝑓(𝑥) =
1
3𝑥−1
𝑥−2
2𝑥 2 −3𝑥+1
𝑥−2
Ex 3: Use the parent graph 𝑓(𝑥) = 𝑥 to graph each function. Describe the transformation(s) that take place.
Identify the new location of each asymptote.
1
a. 𝑔(𝑥) = 𝑥+5
1
b. ℎ(𝑥) = − 2𝑥
4
c. 𝑘(𝑥) = 𝑥−3
6
d. 𝑚(𝑥) = − 𝑥+2 − 4