Preview Notes. Systems of Equations. Linear Functions. Let y = y. Solve for x then solve for y

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1 Preview Notes Linear Functions A linear function is a straight line that has a slope (m) and a y-intercept (b). Systems of Equations 1. Comparison Method Let y = y x1 y1 x2 y2 Solve for x then solve for y 2. Substitution Method Solve for x then solve for y 3. Elimination Method Preview Notes 1

2 Inequalities Words and Inequalities : One Variable Graphing Inequalities: The Half-Plane Words and Inequalities : Two Variable An elevator has a maximum capacity of 12 passengers. True! Let x : number of women y : number of men There are at least twice as many women as men. Let x : number of women y : number of men There are at most 5 more women than men. Let x : number of women y : number of men True! 2 Preview Notes

3 Graphing a Polygon of Constraints The profit per adult is $5 and the profit per child is $2. Maximum Profit = 5x + 2y Test each vertex to see which one gives the largest profit. Finding the Optimal Value The maximum profit of $36 can be made from 4 adults and 8 children. If the maximum (or minimum) value occurs at more than one vertex, it means that every reasonable point between these two vertices will also give maximums (or minimums). We will look at these types of solutions in the practice exams. Let x : Number of Adults y : Number of Children x + y 12 x + y 6 y 2x Max of 12 people Min of 6 people At least twice as many children as adults Preview Notes 3

4 Absolute Value Function Write as equation. Make a number line. Solving Absolute Value Inequalities Test a value between 5 and 1. We will use 3. Recall that... 4 Preview Notes

5 Square Root Functions The Rational Function If a > 0, the graph goes up from the vertex. If a < 0, the graph goes down from the vertex. If b > 0, the graph moves right from the vertex. If b < 0, the graph moves left from the vertex. Finding the Rule of a Square Root Function h is the vertical asymptote. k is the horizontal asymptote. If a > 0 both branches of the function are decreasing. If a < 0 both branches of the function are increasing. Given: the vertex and a point. Preview Notes 5

6 General Form of a Rational Function On a coordinate plane it is obvious that the inverse is a reflection of the original function across the line y = x. To find the asymptotes, use the following functions. Vertical Asymptote Horizontal Asymptote Properties of Inverses The domain of F 1 is the range of F. The range of F 1 is the domain of F. The inverse of a function is not always a function. F F 1 = x and F 1 F = x Inverse of a Function (f 1 ) Given any function F, three steps are needed to find its inverse F Switch the x and y. 2. Solve for y. 3. Restrict the domain of F 1 when working with the absolute value and square root function. 6 Preview Notes

7 Compositions of Functions The composition of two functions is really a function within a function. Solving Exponential Equations Given a x = a y, then x = y. If the bases are congruent, then the exponents are equal. Examples: Solve for x. Preview Notes 7

8 The Transformed Exponential Function By using the exponent laws, the parameters b and h can be eliminated. Note that c > 0 and c 1. k is the horizontal asymptote. Compound Interest There are 4 possible graphs for exponential functions in the form y = a(c) x + k. Note that if a > 0, the function is graphed above the horizontal asymptote, y = k. If a < 0, it is below y = k. Finding the Rule Given the table of values below and the knowledge that k = 3, determine the rule of the exponential function in the form y = a(c) x + k. A(t) = The amount of money you have after t years. P = The principal (amount originally invested) r = The interest rate per year. Always in decimal form, that is (8% = 0.08). n = The number of times the interest is calculated and compounded per year. t = Time (almost always in years) The ± inside the bracket is there because the investment may increase or decrease in value. Use + if the value increases and if the value decreases. 8 Preview Notes

9 Exponential Growth, Decay and Half-Life Q(t) = Quantity at time, t. a = Initial quantity (the amount you start with) r = Rate of growth or decay (sometimes given as a percentage). Note: if something is doubling, 1 + r = 2. If something is increasing by 5%, then 1 + r = = 1.05 t = Time (years, months, days, hours, minutes, or seconds) b = Compound rate (how often the growth or decay happens per unit of time, t) Sample Problem 1 Note: If time is being measured in minutes and the event happens 3 times per minute, then b = 3. If time is being measured in years and the 1 event happens every 500 years, then b = 500. The Laws of Logarithms Preview Notes 9

10 Sample Problem 2 The Graph of the Logarithmic Function The Transformed Logarithmic Function Recall that the exponential function has the rule: The rule for the logarithmic function is the inverse: The actual rule for the transformed logarithmic function is: 10 Preview Notes

11 The Unit Circle Trigonometric Identities In 45 Increments In 30 Increments Preview Notes 11

12 Solving Trigonometric Equations What are the exact values of x that satisfy the equation: Graphing Sine 12 Preview Notes

13 Graphing Cosine Graphing Tangent The only difference between the sine and cosine functions is where we plot our first coordinate, and which direction we move from there. If (a)(b) > 0, start at (h, max) and move down and right. If (a)(b) < 0, start at (h, min) and move up and right. (h, k) is the center or the point of inflection. If (a)(b) > 0, the curved lines are increasing between the vertical asymptotes. If (a)(b) < 0, the curved lines are decreasing between the vertical asymptotes. Preview Notes 13

14 The slope of the tangent line is the negative reciprocal of the perpendicular line back to the origin from the point of tangency. Conics: Ellipses Conics: Circles The equation of a circle centered at the origin is 14 Preview Notes

15 Conics: Hyperbolas Definition of an Ellipse: The sum of the distances from any point on the ellipse to the foci is equal to the length of the major axis. Preview Notes 15

16 Conics: Parabolas Definition of a Hyperbola: The set of points such that the difference of the distances to the foci is constant and is equal to the distance between the vertices. 16 Preview Notes

17 Vectors Vectors have magnitude and direction. Scalars have only magnitude. The distance between the focus and a point on the parabola is the same as the distance between that point and the directrix, regardless of the location of the point on the parabola. Piecewise Functions These types of functions are simply several different functions strung together and defined by their domains (x-values). The rules and domains are given for each function. Preview Notes 17

18 Scalar Product The Acute Angle of Orientation Where θ is the acute angle of orientation and x and y are the components of the vector. The actual angle of orientation depends on the signs of the components. Linear Combinations C is the result of the linear combination of a and b where m and n are scalars. If a and b form a vector basis, then a and b are linearly independent. Two vectors are linearly independent when they are not collinear; that is, the slopes of the two vectors are different. If the angle between two vectors is 0 or 180, the vectors are collinear and not linearly independent. Vector Operations U and V are collinear and are not linearly independent. W and U or W and V are linearly independent and can form linear combinations to form a new vector. 18 Preview Notes