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.

- Example: Evaluating Numerical Expressions

Evaluate `3+4*5`, `(3+4)*5`, `(2+12)/(6-4)`, `(-4)^4`

- 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))`

- 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)`

- 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

- 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.

- 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?

- 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?

- 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?

- 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.

- 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.

- 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)

- 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)`

- 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`

- 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`

- 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?

- 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.

- 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?

- 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

- 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`

- 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`

- 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`

- 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.

- 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?

- Brainstorming: Solving Inequalities Involving Absolute Value

Solve `|2x+4|<=3`, `|3x-1|> 5`

- 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)

- 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)`

- 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)`.

- 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

- 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)`.

- 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`.

- 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`

- 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)`.

- 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.