Math 115 Chapter 1, Section 5 Handout
More on Slope
Slope and Parallel Lines
1) If two nonvertical lines are parallel, then they have the same slope. m1 = m2 .
2) If two distinct nonvertical lines have the same slope, then they are parallel.
The lines don
Math 115 Chapter 1, Section 1 Handout
Graphs and Graphing Utilities
Cartesian coordinate system
Many applications of math involve linear equations and their graphs (straight lines).
We then need to begin with a review of the Cartesian coordinate system.
y
Math 115 Chapter 1, Section 4 Handout
Linear Functions and Slope
Scatter Plot and Regression Line
A scatter plot is a set of points that represent data in a visual form.
A regression line is a line that best fits the data points.
Slope
The slope of the li
Math 115 Chapter 1, Section 2 Handout
Basics of Functions and Their Graphs
Relations
A relation is any set of ordered pairs. The set of all first components of the ordered
pairs is called the domain of the relation and the set of all second components is
Math 115 Chapter 2, Section 8 Handout
Modeling Using Variation
Direct Variation
If a variable y varies directly as a variable x, then we can describe the situation by y = kx,
where k is a nonzero constant called the constant of variation or the constant o
Section 6.1 The Law of Sines The Law of Sines and Its Derivation An oblique triangle is a triangle that does not contain a right angle. An oblique triangle has either three acute angles or two acute angles and one obtuse angle. The angles are labeled A, B
Section 5.5 Trigonometric Equations A trigonometric equation is an equation that contains a trigonometric expression with a variable, such as sin x. In this section, we look at trigonometric equations that are true for only some values of the variable. Th
Section 1.1 Graphs
*Rectangular Coordinate System (a.k.a Cartesian Coordinate System)
Quadrant II
Quadrant I
x-axis y-axis
Quadrant III
Quadrant IV
origin (0,0)
x-axis horizontal number line y-axis vertical number line origin zero points; point of interse
7.1 Systems of Linear Equations in Two Variables Two equations both of which can be written in the form Ax + By = C are called a system of linear equations or a linear system. A solution to a system of linear equations is an ordered pair that satisfies al
Math 115 Precalculus
Review Sheet for Test 1
Section 1.1
Graphing on the Cartesian coordinate system.
Utilizing quadrants I, II, III, and IV and the characteristics of each quadrant.
Find the x and y intercepts.
Section 1.2
Determine whether a relation is
Math 115 Precalculus
Review Sheet for Test 2
Section 7.1
Solving systems linear equations in two variables by graphing, by substitution, and by addition.
Mixture problems. Cost function C ( x ) = cx + f . Revenue function R( x ) = sx .
Profit function P(
Math 115 Chapter 2, Section 4 Handout
Dividing Polynomials; Remainder and Factor Theorems
Long Division of Polynomials
1) Arrange the terms of both the dividend and the divisor in descending order.
2) Divide the first term in the dividend by the first ter
Math 115 Chapter 3, Section 1 Handout
Exponential Functions
Exponential Function
The exponential function f with base b is defined by or y = b x , where b is a positive
constant other than 1 and x is any real number.
Characteristics of Exponential Functio
Math 115 Chapter 2, Section 3 Handout
Polynomial Functions and Their Graphs
Polynomial Function
Let n be a nonnegative integer (no fractions, no variables in the denominator) and let
be real numbers with a n 0 . The function defined by
n
n 1
f ( x ) = a n
Math 115 Chapter 2, Section 6 Handout
Rational Functions and Their Graphs
Rational Functions
p( x )
Rational functions are quotients of polynomial functions. f ( x ) = q( x ) , where p and q
are polynomials with q( x ) 0 . The domain of a rational functio
Math 115 Chapter 2, Section 7 Handout
Polynomial and Rational Inequalities
Polynomial Inequality
A polynomial inequality can be written in one of the following forms
f ( x ) < 0, f ( x ) > 0, f ( x ) 0, f ( x ) 0 where f is a polynomial function
Quadratic
Math 115 Chapter 2, Section 5 Handout
Zeros of Polynomial Functions
Rational Zero Theorem (a.k.a. Possible Rational Roots)
If f ( x ) = a n x n + a n1 x n1 + + a 2 x 2 + a1 x + a0 where the coefficients are integers, the
p
possible rational roots of f ( x
Math 115 Chapter 2, Section 2 Handout
Quadratic Functions
Quadratic Function
Any function of the form f ( x ) = ax 2 + bx + c . Graphs of a quadratic function are known as
parabolas. If the coefficient of x 2 , a, is positive the parabola opens up. If the
Math 115 Precalculus
Review Sheet for Test 3
Section 2.7
Solving polynomial and rational inequalities. Find the boundary points and build a sign table in order
to find the intervals where it works. The position function for a free-falling object near the
Math 115 Chapter 1, Section 3 Handout
More on Functions and Their Graphs
Increasing, Decreasing, and Constant Functions
A function is increasing on an open interval, I, if f ( x1 ) < f ( x2 ) whenever x1 < x2 for any x1 and
x2 in the interval.
A function
Section 2.3 Polynomial Functions and Their Graphs Definition of a Polynomial Function: Let n be a nonnegative integer (no fractions, no variables in the denominator) and let be real numbers with a n 0 . n n -1 2 The function defined by f ( x) = a n x + a
Section 7.2 Systems of Linear Equations in Three Variables Three equations all of which can be written in the form Ax + By + Cz = D are called a system of linear equations in three variables or a linear system in three variables. A solution to a system of
Section 4.5 Graphs of Sine and Cosine Functions In this section, we look at the graphs of sine and cosine functions. We use the traditional symbol x, rather than or t, to represent the independent variable. We use the symbol y for the dependent variable,
Section 4.4 Trigonometric Functions of Any Angle In the last section, we evaluated trigonometric functions of acute angles. The point P=(x,y) was a point r units from the origin on the terminal side of . A right triangle was formed by drawing a line segme
Section 4.1 Angles and Radian Measure ANGLES A ray is a part of a line that has only one endpoint and extends forever if the opposite direction. An angle is formed by two rays that have a common endpoint. One ray is called the initial side (where the meas
3.4 Exponential and Logarithmic Equations exponential equation equation containing a variable in an exponent. Ex: Solving Exponential Equations by Expressing Each Side as a Power of the Same Base If b M = b N , then M = N. 1.) Rewrite the equation in the
Section 2.8 Modeling Using Variation I. Direct Variation If a variable y varies directly as (or is directly proportional to) a variable x, then we can describe the situation by y = kx , where k is a nonzero constant called the constant of variation or the
Section 2.6 Rational Functions and Their Graphs p ( x) Rational Functions are quotients of polynomial function. f ( x) = , where p and q are polynomials q( x) with q ( x) 0 . domain of rational function set of all real numbers except the x-values that mak
Section 2.1 Complex Numbers The imaginary unit, , is defined as , where Ex .
- 7 = -1 7 = i 7
- 16 = - 1 16 = 4i
*Complex Numbers: can be written in the form , referred to as standard form
a = "real part"
b = "imaginary part"
Ex. 3 + i
5i = 0 + 5i
6 = 6 +