College Algebra Video Series

Section numbers refer to Algebra and Trigonometry, Ninth Edition, by Sullivan.

All videos by James Hamblin, Shippensburg University Mathematics Department.

Overview

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

• Example: Evaluating Numerical Expressions
  Evaluate `3+4*5`, `(3+4)*5`, `(2+12)/(6-4)`, `(-4)^4`

Section R.7: Rational Expressions

• Example: Adding and Subtracting Rational Expressions
  Simplify `1/6+3/4`, `2/(x+5)-5/(x-5)`, `2/x+3/(x(x-1))`
• Example: Simplifying Complex Rational Expressions
  Simplify `(1+1/x)/(1-1/x)`, `((4x+1)*5-(5x-2)*4)/(5x-2)^2`
• Example: How to simplify algebraic fractions (and how not to!)
  Simplify `120/104`, `(x(x+4))/(x^2)`, `x^2/(x^2+x^2(x-3))`

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

• Brainstorming: Intercepts and Reading Graphs
  Explain that `(4,3)` is on the graph of `f` iff `f(4)=3`, reading domain/range/intercepts from graph,
• Example: Answering Questions from a Graph 1
  Given a graph, identify intercepts, compute `f(4)`, determine if `f(7)` is positive or negative, and how many solutions `f(x)=1` has

Section 3.3: Properties of Functions

• Brainstorming: Increasing, Decreasing, and Extrema
  Identify from a graph when a function is increasing/decreasing, and locate local and absolute extrema
• Brainstorming: Average Rate of Change
  Explain `ARC = (f(b)-f(a))/(b-a)`, find average rate of change for `f(x)=4x-x^2` from `x=1` to `x=2` and from `x=-1` to `x=6`
• Example: Answering Questions from a Graph 2
  Given a graph, identify intervals when function is increasing/decreasing, find local extrema, compute average rate of change on a given interval
• Example: Prediction and Interpolation
  In 2002, 35.9% of tax returns were files electronically. In 2008, it was 59.8%. (a) Compute the average rate of change from 2002 to 2008. (b) Predict what % of tax returns will be e-filed in 2015. (c) Estimate what % of tax returns were e-filed in 2006.

Section 1.1: Linear Equations

• Example: Simple Linear Equations
  Solve `5x+6 = -18-x`
• Example: Equations with Decimals/Fractions
  Solve `1/2+2/x=3/4`
• Example: Linear Equation Applications
  Sandra is paid time and a half for hours worked in excess of 40 hours. She had gross weekly wages of $442 last week for 48 hours of work. What is her regular hourly rate?

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

• Brainstorming: Distinguishing Between Linear and Nonlinear Relations
  Use scatter plots to determine if data are related linearly or not.
• Brainstorming: Finding a Model for Linearly Related Data
  Given roughly linear data, find a linear model by choosing a good pair of points (does not refer to least-squares or technology)

Section R.4: Polynomials

• Memory Lane: The Parts of a Polynomial
  Define: degree, leading coefficient, leading term, constant term, missing coefficients (zero coefficients). Followed by a simple example of each.
• Memory Lane: Adding, Subtracting, and Multiplying Polynomials
  Add: `(6x^4+x^3-4x^2+5)+(5x^4-x^3+7x)`
  Subtract: `(x^2-3x-4)-(x^3-3x^2+x+5)`
  Multiply: `4x^2(x^3-x+2)`
  Multiply: `(2x-3)(x^3+x+1)`

Section R.5: Factoring Polynomials

• Memory Lane: Factoring Second-Degree Polynomials
  Factor: `4x^5-2x^4+6x^2` (pull out common factor), `x^2-7x-30`, `x^2-8x+12`, `x^2-36`, `n^4-81`

Section 1.2: Quadratic Equations

• Example: Solving Equations by Factoring
  Solve: `x^2-9x=0`, `x^2+4x=21`, `x^3+4x^2+4x=0`
• Example: The Square-Root Method
  Solve: `(2x-5)^2=9`, `(6-x)^2=16`
• Example: The Quadratic Formula
  Solve: `6x^2-7x-3=0`, `2x^2+8=5x`

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

• Brainstorming: The Vertex and Shape of a Parabola
  If `y = ax^2+bx+c`, explain how a affects the graph. Explain the vertex (`-b/(2a)`).
• Brainstorming: The Symmetry of a Parabola
  Suppose `f(x)` is a quadratic function whose vertex is `(4,-3)`. Suppose also that `f(1)=6`. Compute `f(7)`.
• Example: Applications with Parabolas
  A tractor company has found that the revenue, in dollars, from sales of riding mowers is given by `R(p)=-1/2 p^2+1900p`, where p is the price. Determine the price that the company should charge to maximize revenue, and then find the maximum revenue.

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

• Brainstorming: Graphing a Polynomial Using Its Roots
  Determine the end behavior of `-2x^3+5x^2+x-12`, graph `y= 3(x-4)(x+2)^3(x-2)^2`
• Brainstorming: Graphing a Polynomial Using a Sign Chart
  Sketch the graph of `f(x)=-3(x+6)^2(x-1)(x-4)^3`

Section 1.4: Radical Equations; Equations Quadratic in Form; Factorable Equations

• Example: Squaring Once
  Solve `sqrt(x-1)=x-7`, `sqrt(10-x)+4=x`
• Example: Squaring Twice
  Solve `sqrt(2x+3)-sqrt(x+2)=2`
• Example: Equations Quadratic in Form
  Solve `(x+2)^2-3(x+2)-4=0`, `x^4+x^2-6=0`

Section 1.6: Equations Involving Absolute Value

• Brainstorming: The Meaning of Absolute Value
  Simplify `|5|`, `|-10|`, `|x^2|`
• Example: Solving an Equation Involving Absolute Value
  Solve `|x-3|=4`, `|x-3|-5=4`

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

• Brainstorming: Solving Inequalities Involving Absolute Value
  Solve `|2x+4|<=3`, `|3x-1|> 5`

Section 4.5: Inequalities Involving Quadratic Functions

• Brainstorming: Solving Inequalities Involving Quadratic Functions
  Solve `x^2-4x-5 >= 0` (with sign chart), `2x^2 < x+10` (using quadratic formula and graphical reasoning)

Section 6.1: Composite Functions

• Brainstorming: Tables and Formulas
  Given `f(x) = x^2-3x+1` and `g(x)=2x+5`, compute `f(g(x))` and `g(f(x))`.
  Given tables of values for `f` and `g`, compute `(f \circ g)(1)` and `(g \circ f)(-1)`.
• Brainstorming: Algebraically Composing Two Functions
  Given `f(x)=x^2` and `g(x)=2x-3`, find a formula for `(f \circ g)(x)` and for `(g \circ f)(x)`

Section 6.2: One-to-One Functions; Inverse Functions

• Brainstorming: One-to-One Functions and Inverse Functions
  Define one-to-one; `f(x)=x^2` is not one-to-one, `f(x)=2x+3` is one-to-one; horizontal line test. Using bubble-and-arrow diagram to understand the inverse of a one-to-one function.
• Brainstorming: Computing and Graphing Inverses
  Let `f(x)=2x+3`. Find a formula for `f^(-1)(x)`. Graph `f^(-1)(x)` given a graph of `f(x)`.

Section 6.3: Exponential Functions

• Memory Lane: Rules of Exponents
  Interpreting zero, negative, and fractional exponents. Review rules: (1) `x^a * x^b = x^(a+b)`, (2) `x^a/x^b = x^(a-b)`, (3) `(x^a)^b = x^(ab)`
• Brainstorming: Solving Simple Exponential Equations
  Solve `2^(2x)=2^(x+3)`, `4^(x-1)=2^(3x+1)`, `(1/3)^(x-1)=9^(3x-2)`
• Brainstorming: Compound Interest and The Number e
  Define and explain formulas for simple interest, compound interest, and continuously compounded interest

Section 6.4: Logarithmic Functions

• Brainstorming: What is a Log Function?
  Define logs by the relation that `log_a(x)=y` if and only if `a^y = x`
• Brainstorming: What Do Log Functions Look Like?
  Sketch the graph of `y = log_2(x)`, discuss graphical properties of logarithms
• Example: Solving Equations Using Logs 1
  Solve `log_3(x)=2`, `log_x(16)=2`. Evaluate `log_2(16)`.

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

• Example: Log Equations
  Solve `log_10(2x)-log_10(x-3)=1`, `3 log_2(x-1)+log_2(4)=5`
• Example: Solving Equations Using Logs 2
  Solve `2^x = 10`, `5*3^(2x)=8`

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

• Brainstorming: Finding the Asymptotes of a Rational Function
  If `(p(x))/(q(x))` is in lowest terms, then there is a vertical asymptote whenever `q(x)=0`. Find the vertical and horizontal asymptotes of `f(x)= (x^2+6x-27)/(x^2-9)`. Graph this function.
• Example: The Intercepts of a Rational Function
  Find the intercepts of `f(x) = (x^2-6x-16)/(4x^3-10x+2)`, `f(x)=(x^2-5x-6)/(x^2-1)`.

Last modified 15 Aug 2013 by JH.