Haberman / Kling
MTH 111
Section I: Functions and Their Graphs
Unit 9: Graph Transformations: Stretches and Compressions
In Units 7 and 8 we studied what happens when we add or subtract a constant on the inside
or outside of a function, and what happens w
Graphing Systems of Inequalities
REVIEW: When graphing inequalities, start by drawing the equation normally. The
equation should divide the coordinate system into two regions. Then pick a point to find
out which region is the correct side for your inequal
An Introduction to Slope
The graph of a linear function is a straight line.
A line has some degree of steepness, or pitch, which algebra calls its slope.
Every line has some degree of steepness, or
pitch. The degree of pitch ranges from
horizontal, or f
Algebra 2
Unit: Exponential and Logarithmic Functions
Section: Graphing Logarithmic Functions and Domain and Range
Review Worksheet
1) Graph the parent function y = logx
2) What effect does a have on the graph?
3) What effect does h have on the graph?
4)
Algebra 2
Unit: Rational Functions
Section: Graphing Rational Functions and Domain and Range
Review Worksheet
1) What does the graph of the parent function y =
1
look like?
x
2) Describe how a changes the graph of the parent function y =
1
.
x
3) Describe
Function Notation and Finding Function Values
Function notation replaces the y = in an equation with f(x) = which shows that
the equation depends on knowing the values to be used for x.
f(x) shows the y-value for the given x-value and shows that the y-v
Kuta Software - Infinite Algebra 2
Name_
Eccentricity of Conic Sections
Date_ Period_
Identify the eccentricity of each.
2
2
2
x
y
1)
=1
16
9
2
x
y
2)
+
=1
4
25
3) x = 2 y 2
2
2
x
y
4)
+
=1
15 45
2
2
2
y
x2 = 1
5)
16
6) x + y = 4
( x + 2) 2
( x + 1) 2
7)
due
Relations and Functions
7-Jul 1.1 Graphing Basics
8-Jul 1.1.1Using the Cartesian System
8-Jul 1.1.2Thinking Visually
1.2Relatonships between Two Points
1.2.1Finding the Distance between Two Points
1.2.2Finding the Second Endpoint of a Segment
1.3R
An Introduction to Inverses
! The multiplicative identity of any number is 1 because when any number multiplies
with 1 the product is the number itself.
! The multiplicative inverse of any number is the value that, if you multiply it with the
given number
Finding the Distance Between Two Points
Distance formula: d =
( x2 x1 ) 2 + ( y2 y1 )2
The distance formula is an application of the Pythagorean Theorem, which states that
the hypotenuse of a right triangle equals the square root of the sum of the other
Graphing Parabolas
Standard form for a parabola: f (x) = a (x - h)2 + k.
Finding the intercepts of a curve can yield useful points in graphing the curve.
Find the vertex first because thats pretty obvious: (h, k).
Determine whether the curve is a posi
Algebra 2
Unit: Geometry
Section: Geometry of Quadrilaterals
Review Worksheet
1) ABCD is a parallelogram.
B
C
E
A
D
a. AB = 17a 4 and CD = 5a + 52. Find the value of a.
b. The measure of angle BAD equals (4b + 50) and the measure of angle DCB equals (11b
Graphing the Greatest Integer Function
The greatest integer function looks for the greatest integer that is less than or equal
to x.
The graph of this function is sometimes referred to as a step-function because of its
appearance.
The greatest integer b
Graphing Piecewise-Defined Functions
Piecewise functions are those that are defined differently on different intervals of the xaxis.
Piecewise functions can have different
shapes from each other in their various
parts.
Each part must be graphed separately
Graphing the Solution Set of a System of Inequalities
Systems of inequalities might divide the coordinate plane into numerous regions.
When graphing them, see where the regions overlap. That is your solution!
Inequalities are sometimes represented in
thre
Graphing Some Important Functions
This lecture gathers some basic equations with their graphs so you can have a visual
image when you run across each of these examples.
Every point on the line matches some x with
the same value for y, c. The line is not
Graphing Quadratics Using Patterns
To
1.
2.
3.
4.
graph any function, follow these steps:
Determine its basic shape centered at the origin.
Determine its sign so you can sketch how it opens
Determine its steepness so you can sketch its shape
Determine its
Graphing a Line Using Point and Slope
Slope:
rise y2 y1
=
= pitch
run x2 x1
A positive slope indicates that a line rises when viewed from left to right.
A negative slope indicates that a line falls when viewed from left to right.
When graphing a line u
Finding the Vertex by Completing the Square
Standard form of a parabola: f (x) = ax2 + bx + c.
Vertex of a parabola: (h,k).
h = -b/2a.
k = f (h).
Standard form for parabola showing vertex: f (x) = a (x - h)2 + k.
Completing the square is one method for
Graphing Equations by Locating Points
! One way to graph a line is to pick some x-values, substitute them into the function to
find the corresponding y-values, and graph the points.
! Whenever you see one variable squared and the other variable not square
Finding the Second Endpoint of a Segment
Midpoint formula: (
x1 + x2 y1 + y2
,
)
2
2
Any points location is given by the ordered pair (x,y) where x measures horizontal
distance from the origin and y measures vertical distance from the origin.
If you are
Algebra 2
Unit: Discrete Mathematics Sequences and Series
Section: Geometric Sequences and Series
Review Worksheet
1) Find r and the next three terms in each sequence.
a. 3, 6, 12.
b. 20, 10, 5
2) Find t1 and r and then find the 10th and 20th terms of the
Example 1.
a) The term a8b4 occurs in the expansion of what binomial? Answer. (a + b)12. The sum of 8 + 4 is 12.
b) In that expansion, what number is the coefficient of a8b4?
Answer. It is the combinatorial number,
12 11 10 9
12
C4 =
= 495
1 2 3 4
Note ag
Finding the Domain and Range of a Function
The domain includes the set of all real numbers that can be used for x.
The range of a function is the collection of all the y-values or output that exist.
Values excluded from the domain include negatives und
Kuta Software - Infinite Algebra 2
Name_
Graphing and Properties of Ellipses
Date_ Period_
Identify the center, vertices, co-vertices, foci, length of the major axis, and length of the minor axis of each.
2
2
2
2
x
y
1)
+
=1
49 169
x
y
2)
+
=1
36 16
x2
y2
Kuta Software - Infinite Algebra 2
Name_
Double- and Half-Angle Identities
Date_ Period_
Use a double-angle identity to find the exact value of each expression.
1) sin 120
3) cos
4
3
2) tan 60
4) sin
5
3
Use a half-angle identity to find the exact value o
Finding the Maximum or Minimum of a Quadratic
Every parabola has a turning point, or vertex. This point is the minimum point on a
positive parabola and the maximum point on a negative point.
The maximum or minimum value relates to the height of the curv
Multiple Choice
1. For some complex number z = r(cos()+i sin(), z 6 = 32z. Which of the following
could be the value of z?
(a) 2(cos 6 + i sin 6 )
2
+
i
sin
)
(b) 2(cos 2
5
5
(c) 2(cos 3 + i sin 3 )
(d) 2(cos 2 + i sin 2 )
(e) None of the above
2. Given t
Multiple Choice
1. For what value of n will the coefficient of the term containing x6 in the binomial
expansion of (2x3 + y 5 )n be equal to 60?
(a) 3
(b) 4
(c) 5
(d) 6
(e) None of the above
2. The amount of carbon-14 in a plant is given by the equation D
Multiple Choice
1. For what value of n will the coefficient of the term containing x6 in the binomial
expansion of (2x3 + y 5 )n be equal to 60?
(a) 3
(b) 4
(c) 5
(d) 6
(e) None of the above
Solution: (d): The term containing x6 in the binomial expansion
Finding the Slope of a Line Given Two Points
Slope is the change from one point to another on a line. It is stated as a ratio (fraction)
with change vertically over change horizontally. This is frequently referred to as
rise/run.
Slope is the measure of p
Kuta Software - Infinite Algebra 2
Name_
Graphing and Properties of Parabolas
Date_ Period_
Identify the vertex, axis of symmetry, and direction of opening of each.
2
1) y = 2( x + 10) + 1
2) y =
1
( x 7) 2 + 1
3
16
46
1
3) y = x 2 +
x
3
3
3
4) y = 2 x 2
Algebra 2
Unit: Geometry
Section: Geometry of Triangles
Review Worksheet
1) Triangle ABC is congruent to triangle XYZ.
Z
A
Y
B
C
X
a. The measure of angle B is 42 degrees and the measure of angle C is 39 degrees. The measure of
angle Y is 5a + 10 degrees.
Kuta Software - Infinite Algebra 2
Name_
Graphing and Properties of Hyperbolas
Date_ Period_
Identify the vertices, foci, and direction of opening of each.
2
2
2
x
y
1)
=1
81
4
2
3)
5)
2
2
y
x
=1
25
16
( x + 2) 2
169
2
x
y
2)
=1
121
81
4)
( y + 8) 2
=1
4