Algebra 2 Graphing Unit
Name:_________________________
Date:_____________ Period______
I. Absolute Value Functions
Absolute Value functions are piecewise functions. They have different functions
for different parts of the domain.
Parent Function: y = │x│ means
y = x when x ≥0
y = –x when x < 0
The vertex, as with a parabola, is (h,k). In this case, the vertex is (0,0), which is
the position where the domain of the function changes.
To graph, plot the vertex then use an x/y chart to find the other points. Use a
minimum of 4 additional points.
Ex: y = │x│
X
Y
Domain: All Real Numbers,
Range: { y: y ≥ 0 } Min. Value is 0.
Absolute value functions with a degree of 1 have a characteristic “V” shape.
The Domain of the entire function is all real numbers, but the vertex divides the
domain so that the two graphs, rays, intersect at (h,k).
Vertex Form:
Graphing Form:
y – k = a│x – h│
y = a│x – h│ + k
Example:
y = │x + 3│ – 6
Vertex: (–3, –6)
To graph: y = │x + 3│ – 6
X
Y
Transformations: Translates Left 3, Down 6
Domain:
Range: { y: y ≥ –6 }
Min. value: –6
Axis of symmetry : x = –3
y-intercept (x = 0): –3
x-intercepts: Solve when y = 0
Example:
X
y = │x – 2│ + 1
Vertex : _________
Y
Transformations: ____________________
Domain: _______
Min. value: _____
Range: ______________
y-intercept: ________
x-intercepts: ___________________
Axis: ________
Reflections: The sign of the coefficient in front of the absolute value symbol
determines a reflection. If the sign is negative, the graph reflects over the xaxis. The V opens down.
Ex. y = –│x│
X
0
1
2
-1
-2
Vertex: (0, 0)
Y
0
-1
-2
-1
-2
Domain:
,
Range: { y: y≤ 0 },
Max Value is 0
DILATIONS: Dilations change the shape of the graph by making it stretch or
compress. A vertical stretch (along the y-axis) will make the graph appear thinner”.
A vertical compression will make the graph appear “wider”.
Example: Graph the following functions on your graphing calculator and sketch the
graphs. Label them A, B, and C.
A) y = │x│
B) y = 2│x│
Which graph shows a stretch? _______
Which graph shows a compression? ________
Domain:
,
Range: {y: y ≥ 0}
Min. = 0
C) y = ½ │x│
EXAMPLE:
y = –4│x + 3│ + 2
Vertex: __________
X
Y
Domain: _______
Range: ______________
Max/Min. value: _________
Axis: ________
Transformations: _________________________________________________
Problems:
Graph the following and list the vertex, axis of symmetry, domain, range, any
transformations and the max/min value. All transformations are in relation to the
parent graph y = │x│.
X
Y
1. y = │x + 3│
Vertex: _______
Domain: _______
Range: ______________
Max/Min. value: _________
Axis: ________
Transformations: _________________________________________________
2. y = │x│ + 3
X
Y
Vertex: _______
Domain: _______
Range: ______________
Max/Min. value: _________
Axis: ________
Transformations: ______________________________________________
3. y = –│x – 2│ + 1
X
Y
Vertex: _______
Domain: _______
Range: ______________
Max/Min. value: _________
Axis: ________
Transformations: ________________________________________________
4. y =
1
│x│
3
X
Y
Vertex: _______
Domain: _______
Range: ______________
Max/Min. value: _________
Axis: ________
Transformations: ________________________________________________
5. Use a graphing calculator to sketch
the following graphs and label them.
A.)
y = │x│
B.)
y = 2│x + 4│ – 2
Fill in the following blanks for graph B.
Vertex: _______
Domain: _______
Range: ______________
Max/Min. value: _________
Axis: ________
Transformations: ________________________________________________
II. Linear Equations:
Parent graph: y = x
Slope-Intercept form: y = mx + b
Graph the following on the same grid. List the domain and range and state the
translations from the parent graph. Mark each line with its capital letter.
A) y = x
Domain: ______
Range: ______
B) y = x + 2
Translations: _________________________
C) y = x – 3
Translations: _________________________
Graph the following on the same grid.
A) y = x
B) y = 2x
C) y = –x
D) y = –x – 2
Special Cases: Graph each of the following on the same grid.
A) x = 3
Domain: ________ Range: ________
B) x = –4
Domain: ________ Range: ________
C)
y=3
D) y = –4
Domain: ________ Range: ________
Domain: ________ Range: ________
III. CUBIC EQUATIONS:
Parent graph: y = x3
The cubic equation is different from a parabola in that as
x goes to negative infinity, y goes to negative infinity.
For a parabola, as x goes to negative infinity, y goes to
positive infinity. The same is true for absolute value
function. The cubic equation goes in both directions,
positive and negative.
Another way of looking at the difference between the
graphs is as follows: As you read from left to right, the graph of a cubic equation,
y = x3, is always increasing. In fact, it is called an “increasing” function. As you
read from left to right, the graph of a quadratic or absolute value equation
decreases until it reaches the vertex and then it beings to increase.
Ex:
X
0
1
-1
2
-2
y = x3
Y
0
1
-1
8
-8
At the point (0, 0), notice that the curve changes. It goes from concave down to
concave up. This point is called an inflection point. We can use this point to
“center” our graphs just like we used the vertex for quadratics.
Domain: All Real Numbers,
Range: All Real Numbers,
Notice that there is no minimum or maximum.
“Vertex” Form:
y − k = a(x − h)3
Graphing Form:
y = a(x − h)3 + k
Inflection Point:
(h, k)
Transformations
The transformations are the same for the cubic graph as they were for the
quadratic or absolute value.
Reflection: If you have y = − x3 then the graph will reflect over the x-axis, which
then has the graph decreasing as you read from left to right.
Example:
Graph the following. Use the x-y chart. Check the graph with a graphing calculator.
A.) y = x3
B.) y = − x3
C.) y =
X
1
(x − 3)3 − 1
3
Y
Describe the transformations:
B.) __________________________________
C.) ____________________________________________________________
For graph C, give the inflection point: ___________
Problems
Graph the following. List the inflection point, domain, range, and any
transformations from the parent graph y = x3. Check your graphs on the graphing
calculator.
1.) y = (x − 2)3
X
Y
Inflection: __________
Domain: _____________
Range: ______________
Transformations: ______________________________________________
2.) y − 2 = x3
X
Y
Inflection: __________
Domain: _____________
Range: ______________
Transformations: ______________________________________________
y = x3
3.)
A) y =
X
Y
X
Y
1 3
x
2
B) y = 2x3
Transformations:
A.) _______________
__________________
B.) ____________________________________
For the next problem, just use the graphing calculator to sketch the graphs. Then
describe the transformations.
4.)
C.)
y = x3
D.)
y=−
1
(x + 1)3 − 1
2
For graph D, give the following information.
Inflection: __________
Domain: _____________
Range: ______________
Transformations: ______________________________________________
Parent Graph: y =
IV. Square Root Functions
x
The square root function is really half of a parabola opening
right or left. This function is written as y = x . The
square root of a negative number is not a real number, so
for this function, x can not be negative. This limits the
domain as well as the range.
In this function, the vertex is also the starting point of the
graph.
EX: y =
x
X
Y
0
0
1
1
4
2
9
3
Vertex: (0, 0)
Domain:
{x: x ≥ 0}
Range: {y: y ≥ 0}
Minimum Value: 0
“Vertex” Form:
Example: y =
No Axis of Symmetry
y −k = a x −h
x+4 − 3
Graphing Form:
X
Y
Vertex: __________
Domain: __________
Range: ___________
Transformations : __________________
y = a x −h + k
Problems
Graph and label each graph. List the vertex, domain, range, and transformations.
1.) A) y =
B) y =
x
x + 3
Vertex: _____________
X
Y
Domain: ____________
Range: _____________
Transformations : _______________
C) y =
x − 4
X
Y
Vertex: _____________
Domain: ____________
Range: _____________
Transformations : _______________
Graph the following on the calculator and sketch the graphs. Label the graphs.
2.) A) y =
x−4
Domain: ________
Vertex: _________
Range: _________
Transformations : _______________
B) y =
x+3
Domain: ________
Vertex: _________
Range: _________
Transformations : _______________
C) y = − x
Vertex: _________
Domain: ________
Range: _________
Transformations : _______________
1
x
Rational Functions are the quotient of two polynomials,
V. Rational Functions
f(x) =
Parent Graph:
y=
g(x)
, h(x) ≠ 0
h(x)
Asymptotes are lines that the graph approaches but never
touches. Rational functions have both vertical and horizontal
asymptotes. For now, the graphs will have only one of
each. There is a vertical asymptote because the denominator, x, can not be zero.
There is a horizontal asymptote because no value of x will make y equal to zero.
Asymptotes are dotted because they are not part of the graph, but help when
creating the graph. The “vertex” or centering point of the graph is the
intersection point of the asymptotes.
Ex: y =
1
x
X
Y
0
///
1
1
2
1/2
4
1/4
Centering Point (h, k): (0, 0)
V. Asym.: x = 0
H. Asym.: y = 0
Domain:
{x: x ≠ 0}
Range: {y: y ≠ 0}
“Vertex” Form:
 1 
y − k = a

 x − h
Graphing Form:
 1 
y = a
 + k
 x − h
Example: y =
1
+ 4
x+2
X
Y
Center: __________
V. Asym.:
__________
H. Asym.: __________
Domain:
__________ Range: __________
Transformations: _________________________
Problems
Graph and list the center, asymptotes, domain, range, and transformations.
1.) y = −
1
x
X
Y
Center: __________
V. Asym.:
__________
H. Asym.: __________
Domain:
__________ Range: __________
Transformations: _________________________
2.) y =
1
x−3
X
Y
Center: __________
V. Asym.:
__________
H. Asym.: __________
Domain:
___________
Range: __________
Transformations: _____________________________
3.) y =
1
− 3
x
X
Y
Center: __________
V. Asym.:
__________
H. Asym.: __________
Domain:
___________
Range: __________
Transformations: _________________________
Graph the following on your calculators. Sketch the graphs and give the requested
information.
4.)
y=
2
x −1
Center: __________
V. Asym.:
Domain:
________
H. Asym.: ________
___________
Range: __________
Transformations: __________________________
5.)
y= −
1
+ 3
x−3
Center: __________
V. Asym.:
Domain:
________
H. Asym.: ________
___________
Range: __________
Transformations: __________________________