1. 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

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

3. 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?

4. Example 3 – continued

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

6. 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?

7. 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 ”)

8. 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

9. Example 1
Evaluate the function at the indicated values.

10. 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

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

12. 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.

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

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

15. 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:

16. Composition
Find the composition function:

17. 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

18. Slope y x

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

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

21. 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

22. 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

23. 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

24. 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.

25. 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.

26. 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

27. Example 1

28. Example 2

29. 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?

30. Example 3 – continued

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

32. 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)

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

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

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

36. 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

37. 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

38. Example 4
Demand:
Supply:

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

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

41. Example 1

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

43. Examples:
x =
x =

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

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

46. x-value of the vertex

47. Example 1

48. Example 2

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

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

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

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

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

54. 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.

55. 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.

56. 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 50
100 x

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

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