College Algebra Video Series
Section numbers refer to Algebra and Trigonometry, Ninth Edition, by Sullivan.
All videos by James Hamblin, Shippensburg University Mathematics Department.
This video series contains three types of videos:
- Memory Lane: These videos remind you of important concepts that you learned in your previous studies.
- Brainstorming: These videos explore new topics and contain important definitions or mathematical facts that we will be using.
- Examples: These videos contain example problems that are worked out, step by step.
Watch each video multiple times if necessary, taking notes where appropriate.
Section R.1: Real Numbers
Section R.7: Rational Expressions
Section 3.1: Functions
- Memory Lane: "Bubble and Arrow" Functions
Example of function: US States -> # of US House Representatives
Example of non-function: People -> Phone number
- Memory Lane: Domain and Range
Examples: `f(x)=x^2`, `f(x)=1/x`, `f(x)=sqrt(x)`
- Example: Evaluating Function Notation
Given `f(x)=(2x-1)/(3x+2)`, compute `f(2)`, `f(-1)`, `f(a+3)`, `2 f(-x)`, `f(x+h)`
- Example: Combining Functions
Given tables of values for `f` and `g`, compute `f(0)`, `(f+g)(-2)`, `(f-g)(2)`, `(f*g)(-1)`, `(f/g)(3)`
Section 3.2: The Graph of a Function
Section 3.3: Properties of Functions
Section 1.1: Linear Equations
Section 1.7: Problem Solving (Linear)
- Example: Interest Problem (Linear)
A bank loaned out $12,000, part of it at the rate of 8% per year, and the rest at the rate of 18% per year. If the interest received totaled $1000, how much was loaned at 8%?
- Example: Mixture Problem (Linear)
A coffee manufacturer wants to market a new blend of coffee that sells for $3.90 per pound by mixing two coffees that sell for $2.75 and $5 per pound, respectively. What amounts of each coffee should be blended to obtain the desired result?
Section 2.1: The Distance and Midpoint Formulas
- Brainstorming: The Cartesian Plane
Define x-axis, y-axis, origin, quadrants, how to plot a point (a,b)
- Brainstorming: The Distance and Midpoint Formulas
Deriving distance and midpoint formulas using geometry
- Example: Applications to Geometry and the Real World
1. Consider the three points A(-2,1), B(2,3), C(3,1). Find the length of each side, and verify that the triangle is a right triangle.
2. A baseball diamond is a square, 90 feet on a side. A shortstop is standing exactly halfway between 2nd and 3rd base, and needs to throw the ball to home plate. How far away from home plate is he?
Section 2.3: Lines
- Memory Lane: The Idea of Slope
Slope is "rise over run", slope of a vertical line, how to "eyeball" the slope from a graph
- Memory Lane: Vertical and Horizontal, Parallel and Perpendicular
Horizontal lines look like y=a, vertical lines look like x=b, parallel lines have equal slopes, perpendicular lines have negative reciprocal slopes
- Brainstorming: The Point-Slope and Slope-Intercept Forms
Deriving the forms from the slope formula
- Example: Applications 1
A cereal company spends $40k on advertising to sell 100k boxes, and $60k to sell 200k boxes. (a) Write a linear equation that relates the amount A spend on ads to the number x of boxes sold. (b) How much needs to be spend to sell 300k boxes? (c) Interpret the slope of your linear function.
- Example: Applications 2
A truck rental company charges $30 to rent a truck for the day, plus an additional charge for mileage. The total cost of renting a truck and driving 100 miles is $65. (a) Find a linear equation that relates the cost C of renting a truck to the number n of miles driven. (b) Bonnie rents a truck, and her total cost was $110.50. How many miles did she drive? (c) Interpret the slope of your linear function.
Section 4.1: Linear Functions and Their Properties
- Brainstorming: Using Average Rate of Change to Identify Linear Functions
Given two different tables of values, compute ARC to determine if the functions are linear or not
- Example: Supply and Demand
Define "quantity supplied", "quantity demanded", and "equilibrium price." Suppose `S(p)=-200+50p` and `D(p)=1000-25p`. (a) Graph these equations. (b) Find the equilibrium price and equilibrium quantity. (c) Determine the prices for which the quantity demanded is higher than the quantity supplied.
Section 4.2: Linear Models; Building Linear Functions from Data
Section R.4: Polynomials
Section R.5: Factoring Polynomials
Section 1.2: Quadratic Equations
Section 1.7: Problem Solving (Quadratic)
- Example: Uniform Motion Problem
A motorboat heads upstream a distance of 24 miles on a river whose current is running at 3 miles per hour. The trip up and back takes 6 hours. Assuming that the motorboat maintained a constant speed relative to the water, what was its speed?
- Example: Another Uniform Motion Problem
A plane normally travels from City A to City B, a distance of 1200 miles, at a speed of 300 miles per hour. One day, the plane encountered a headwind, and the trip took an extra hour. What was the speed of the headwind?
- Example: "Working Together" Problem
It takes Andy 40 minutes to do a particular job alone. It takes Brenda 50 minutes to do the same job alone. How long would it take them if they worked together?
- Example: Quadratic Word Problem
If Eli and Payton work together to clean their house, it takes them 2 hours. Working alone, Payton can do it three hours faster than Eli can. How fast can Payton clean the house?
Section 4.3: Quadratic Functions and Their Properties
Section 4.4: Build Quadratic Models from Verbal Descriptions and From Data
- Example: Building a Quadratic Model
A farmer with 4000 meters of fencing wants to enclose a rectangular plot that borders on a river. If the farmer does not fence the side along the river, what is the largest area that can be enclosed?
Section 3.4: Library of Functions
- Memory Lane: Library of Common Functions
Basic properties (domain, intercepts, symmetry) of constant function, identity function, square function, cube function, square root, cube root, reciprocal function, and absolute value
Section 5.1: Polynomial Functions and Models
Section 1.4: Radical Equations; Equations Quadratic in Form; Factorable Equations
Section 1.6: Equations Involving Absolute Value
Section 3.4: Piecewise-Defined Functions
- Brainstorming: Piecewise-defined Functions
Given `f(x)=x^2` if `x<2`, 5 if `x=2`, and `-2x+10` if `x>2`, evaluate `f(-3)`, `f(4)`, `f(2)`, and graph `y=f(x)`
- Example: Application of Piecewise-Defined Functions
Friendly Energy Company charges a $15.95 monthly service charge, plus $0.33 per therm for the first 50 therms, and $0.27 per therm over 50. (a) What is the charge for using 50 therms in a month? (b) What is the charge for using 150 therms in a month? (c) Find a piecewise model for the monthly charge C as a function of the number of therms x.
Section 1.5: Solving Inequalities
- Brainstorming: Interval and Inequality Notation
Write various intervals (expressed as simple and compound inequalities) in interval notation
- Brainstorming: Solving Inequalities
Solve `x-5>8`, `4-3x<=19`, `4<2x-1<9`
- Example: Simple Inequalities
What is the domain of `sqrt(8-2x)`?
Annie's monthly electric bill is $35 plus 9.44 cents per kilowatt-hour used that month. If Annie wants to make sure that her bill is no more than $60, how many kilowatt-hours can she use?
- Example: Compound Inequalities
Solve `-3 <= 3-2x <=12`.
Sally the car saleswoman receives a commission for every car she sells, equal to $200 plus 40% of the selling price. She expects a particular car to sell for between $4,000 and $5,500. What commission should she expect to receive?
Section 1.6: Inequalities Involving Absolute Value
Section 4.5: Inequalities Involving Quadratic Functions
Section 6.1: Composite Functions
Section 6.2: One-to-One Functions; Inverse Functions
Section 6.3: Exponential Functions
Section 6.4: Logarithmic Functions
Section 6.5: Properties of Logarithms
- Example: Using Log Properties
Discuss basic properties of logs and their exponential counterparts. Write `log_2(x*sqrt(x^2+1))` as a sum of logarithms. If `log_b 2 = X` and `log_b 3 = Y`, write `log_b 12` in terms of `X` and `Y`.
Section 6.6: Logarithmic and Exponential Equations
Section 5.2: Properties of Rational Functions
- Brainstorming: Finding the Domain of a Rational Function
Domain is all real numbers except those that make the denominator equal zero. Find the domain of `f(x) = (x^2+x-5)/(x^2-x-6)`. Find the domain of `f(x)=(3x^3-x^2+16)/((x-3)^2(x+5)(x+1))`.
- Brainstorming: What is an Asymptote?
Define vertical asymptote. Use table of values to show that `f(x)=(2x)/(x-3)` has a vertical asymptote at `x=3`. Define horizontal asymptote. Use table of values of show that `f(x)=(2x)/(x-3)` has a horizontal asymptote at `y=2`. Graph `f(x)=(2x)/(x-3)`.
Section 5.3: The Graph of a Rational Function
15 Aug 2013