Solving Linear Equations Solving equation = finding value(s) for the unknown that make the equation true Add, subtract, multiply, divide, raise to powers, take roots, find least common denominators to isolate the unknown

Example 1: solve for x Example 2: solve for t

Example 3 Invest $140,000 with the goal of collecting $10,000 in interest each year. How much should go in risky account earning 10% interest and how much should go into safer account earning 5% interest?

Example 3 – continued

Example 4 If profit is linked to number of items sold by the following equation, where will the company reach $60,000?

Example 5 When a $980,000 building depreciates in value for tax purposes, its value, y, after x months of use is given by: How many months will it take for the building to fully depreciate? How many years?

Functions One quantity depends on another quantity # of shirts => revenue Age => Height Formal definition: A function f is a rule that assigns to each element x in a set A exactly one element, f(x), in a set B. ( f(x) is read as “ f of x ”)

Algebraic & Graphic Algebraic: Graphic: Plot points (x, f(x)) in the two dimensional plane f(x) x -2 5 9 -4 1 f(x) 41 42 -3 27 42 21 x 2 4 6 8 -21

Example 1 Evaluate the function at the indicated values.

Domain Cannot divide by zero Cannot have negatives inside an even root The set of all inputs for which the function is a real number. Three restrictions on the domain for any function: Cannot divide by zero Cannot have negatives inside an even root Cannot have zero or negatives inside logarithms

Examples Find the domain for the following functions. D: D:

Graphs of Functions Functions in one variable can be represented by a graph. Each ordered pair (x, f(x)) that makes the equation true is a point on the graph. Graph function by plotting points and then connecting the points with smooth curves.

Example y 6 Create a table of points: x -3 -2 -1 1 x -1 1 -6

Domain/Range from Graph Look at graph to determine domain(inputs) and range (outputs). f(x) Domain: 2 Range: -2 2 4 x -2

Combining Functions Given the functions: Take simple functions and combine for more complicated ones Arithmetic - add, subtract, multiply, divide Composition – evaluate one function inside another Given the functions: Domain: Domain: Domain: Domain:

Composition Find the composition function:

Linear Functions Linear functions also known as lines. Each line is defined by: intercepts and slope Slope is the change in y over the change in x rise over run

Slope y x

Forms of Line General equation: Ax + By + C = 0 Slope-intercept equation: slope = m, y-intercept = b Point-slope equation: slope = m, point =

Example 1 Given two points, (-2, 9) and (4, 1), on a line, find the equation of the line. 9 -2 4

Example 2 The U.S. population(in millions) can be described as a function of time where the independent variable is the number of years past 1990 with: Find p(25) and explain what it means. Graph the function. 750 500 10 20

Example 3 Suppose the cost of a building is $960,000 and a company wants to use a straight-line depreciation schedule for 240 months. Write the equation for this depreciation schedule. y = value of building, x = months

Systems of Linear Equations A set of equations involving the same variables A solution is a collection of values that makes each equation true. Solving a system = finding all solutions

Substitution Method Pick one equation and solve for one variable in terms of the other. Substitute that expression for the variable in the other equation. Solve the new equation for the single variable and use that value to find the value of the remaining variable.

Elimination Method Multiply both equations by constants so that one variable has coefficients that add to zero. Add the equations together to eliminate that variable. Solve the new equation for the single variable and use that value to find the value of the remaining variable.

Equivalent Systems of Linear Equations Swap the position of two equations Multiply equation by non-zero constant Add a multiple of one equation to another equation Use Left-to-Right Elimination and then Backward Substitution

Example 1

Example 2

Example 3 Each serving (1 cup) of milk contains 430 mg of potassium and 2.5 g of fat. Each serving (100 g) of peanuts contains 690 mg of potassium and 45 g of fat. If a diet needs to contain 1205 mg of potassium and 27.5 g of fat, then how much of each food is needed?

Example 3 – continued

Applications with Linear Functions Cost, revenue, profit Marginals for linear functions Break Even points Supply and Demand Equilibrium

Cost, Revenue, Profit, Marginals Cost: C(x) = variable costs + fixed costs Revenue: R(x) = (price)(# sold) Profit: P(x) = R(x) – C(x) Marginals: what would happen if one more item were produced (for marginal cost) and sold (for marginal revenue or marginal profit)

Example 1 Find C(50), R(50), P(50) and interpret. Find all marginals when x = 50 and interpret.

Example 1 – continued Find all marginals when x = 50 and interpret. For linear functions, the marginals are the slopes of the lines.

Break Even Points Example 2 Companies break even when costs = revenues or when profit = 0. Example 2

Example 3 If P(10) = -150 and P(50) = 450, how many units are needed to break even if the profit function is linear? The company breaks even by producing and selling

Law of Demand: quantity demanded goes up as price goes down Law of Demand: quantity demanded goes up as price goes down. Likewise, as price goes up, quantity demanded goes down. Law of Supply: quantity supplied goes up as price goes up. Likewise, as price goes down, quantity supplied goes down. Market Equilibrium: where quantity demanded equals quantity supplied

Example 4 Demand: Supply:

Quadratic Equations Equations involving three terms constant, linear, and square Take square root, factor or use quadratic formula

Factoring Since AB = 0 implies either A=0 or B = 0, then rewrite the quadratic equation as a product of linear factors.

Example 1

Quadratic Formula x = If factoring not nice, remember quadratic formula: For given quadratic equation, find solutions for x with: x =

Examples: x = x =

Application A rectangular garden is 8 feet longer than it is wide. If its area is 240 square feet, then what are its dimensions?

Quadratic Functions & Applications Graph is a parabola. Either has a minimum or maximum point. That point is called a vertex and is

x-value of the vertex

Example 1

Example 2

Applications Example 3 Find the break-even point for the given cost and revenue functions. Company breaks even if it produces and sells

Example 4 Find the maximum revenue for the revenue function given. Must sell units in order to maximize revenue. Maximum revenue is $

Example 5 Find max revenue where the demand for a product is given. Must sell units in order to maximize revenue. Maximum revenue is

Example 6 Equilibrium point: Find the equilibrium point for the given demand and supply functions. Equilibrium point:

Rational Functions p(x) and q(x) are polynomials, with

Main Characteristic: Asymptotes Asymptotes are lines (horizontal or vertical) that the function approaches for certain inputs. Horizontal: for larger and larger values of x Vertical: for inputs that get closer to a domain restriction, in this case, a value that causes division by zero.

Asymptotes Horizontal: y = k is a horizontal asymptote when n < = m. Vertical: x = c is a vertical asymptote when p(c) does not equal 0 but q(c) does equal 0.

Example 1 Asymptotes: Create a table of points: f(x) x x f(x) -1.5 1 -1.5 1 -7 1.5 -18 1.9 -106 2.5 26 3 15 10 5.375 20 4.611111 50 4.229167 100 4.112245 x

Example 2 Asymptotes: Find a few points: f(x) x f(x) -1 -6 2 20 3 15 x

Example 3 If the cost function for producing x items is given, describe the associated average cost function. Asymptotes: x