1.5 Solving Inequalities Use Interval Notation Example 1 Example 2 Solve Inequalities Example 3 1105 11e Notes Page 1 Example 4 Solving a Combined Inequality Solve: Example 5 Example 6 1105 11e Notes Page 2 Example 7 1105 11e Notes Page 3 1.6 Equations and Inequalities Involving Absolute Value Solve Equations and Inequalities Involving Absolute Value Example 1 a) |5x| = 15 d) b) |3 - 8t|-3 = 11 c) |2|x = -16 e) Example 2 Solve Each Inequalitity. Graph the solution set. a) |1 - 4x| - 9 < -6 b) |9 - 2x|-2 > 11 1105 11e Notes Page 4 c) d) |-3x| > |-11| e) 4|7x| ≥ -4 1105 11e Notes Page 5 2.1 The Distance and Midpoint Formula Distance Formula: Midpoint Formula: Example 1: Find the distance d between the points (-4, 5) and (3, 2). Example 2 Find the distance between the points (4, -3) and (2, 5). Example 3: Consider the three points A=(-2, 1), B=(2, 3), and C= (3, 1). a) Plot each point and form the triangle. b) Find the length of each side of the triangle. c) Verify that the triangle is a right triangle. d) Find the area of the triangle. 1105 11e Notes Page 6 Example 4 If (4, -2) is the endpoint of a line segment, and (-9, -1) is its midpoint, find the other endpoint. Example 5 Find the midpoint of the line segment from to Example 6 A Car and a Truck leave an intersection at the same time. The car heads east at 30mph, while the truck heads south at 40mph. Find an expression for the distance apart d (in miles) at the end of t hours. 1105 11e Notes Page 7 2.2 Intercepts; Symmetry Find Intercepts To find the intercepts of an equation: x-intercept: Make y=0, solve, (#, 0) y-intercepts: Make x=0, solve, (0, #) Example 1: Find the intercept(s) of the graph Test an Equation for Symmetry 1105 11e Notes Page 8 Example 2: Test for symmetry. 1105 11e Notes Page 9 Example 3: Test for symmetry. Example 4: Given the equation, test for symmetry. 1105 11e Notes Page 10 2.3 Lines Objective 1: Calculate and Interpret the Slope of a Line Example 1: Find and interpret the slope of the line containing the points (1, 2) and (5, -3). Objective 2: Find the Equation of the Line Given Two Points When you see FIND, EQAUTION, LINE, follow these steps: 1. Find the slope 2. Using slope-intercept form y=mx+b, plug m (the slope), x and y from a given point to get b. 3. Once you have b plug it into y=mx+b along with m (the slope). 4. If needed Standard/General Form: Ax+By=C (fractions are not allowed and A must be positive). Example 2: Find an equation of the line containing the points (2, 3) and (-4, 5). Express in Standard Form. - Parallel Lines: Have the same slope, but different y-intercept. - Perpendicular Lines: Have negative reciprocal slopes, meaning you flip and change the sign. The product of perpendicular slopes = -1 - If the equation is the same, the lines are the same. Example 3: Find an equation of the line that contains the point (1, -2) and is perpendicular to the line x+3y=6. (b) Find the line containing the same point and parallel to the same line. 1105 11e Notes Page 11 Undefined Slopes x=# has always an undefined slope. It creates a vertical line. Zero Slope y=# has a slope of 0. It creates a horizontal line. Example 4 Graph the line that goes through (2, -3) and has an undefined slope. Find the equation of the line. Example 5 Graph the line that goes through (-1, 9) and has a slope of 0. Find the equation of the line. Example 6 Find the equation of a line that is parallel to the line x=11 and contains the point (-2, 7). Example 7 Graph 3x-2y=6 To graph using a ti-84: 1) Press y= (top left) 2) Make sure the equation has isolated the y and input the equation into y1= 3) Press graph 4) To get points (table), press 2nd, then graph. 1105 11e Notes Page 12 2.4 Circles General Form of a Circle: Everything is foiled from standard form. The Unit Circle: (used in calculus) Example 1: Write the standard form of the equation of the circle with radius 5 and center (-3, 6). Example 2: Graph the equation: Example 3: For the circle , find the intercepts, if any, of its graph. 1105 11e Notes Page 13 Example 4: Graph the equation: Example 5: Graph . 1105 11e Notes Page 14 Example 6 A Ferris wheel has a maximum height of 331 feet and a wheel diameter of 310 feet. Find an equation for the wheel if the center of the wheel is on the y-axis and y represents the height above the ground. 1105 11e Notes Page 15 2.5 Variation Varies Directly, Directly Proportional, and Constant of Proportionality If y varies directly with x, x > 0, and k, k > 0, is the constant of proportionality, then the graph illustrates the relationship between y and x. Note that the constant of proportionality is, in fact, the slope of the line. Example 1: Mortgage Payments The monthly payment p on a mortgage varies directly with the amount borrowed B. If the monthly payment on a 30-year mortgage is $6.25 for every $1000 borrowed, find a formula that relates the monthly payment p to the amount borrowed B for a mortgage with these terms. Then find the monthly payment p when the amount borrowed B is $200,000. The graph illustrates the relationship between the monthly payment p and the amount borrowed B. 1105 11e Notes Page 16 Varies Inversely and Inversely Proportional The graph illustrates the relationship between y and x if y varies inversely with x and k > 0, x > 0. Example 2: Maximum Weight That Can Be Supported by a Piece of Pine The maximum weight W that can be safely supported by a 2-inch by 4-inch piece of pine varies inversely with its length l. Experiments indicate that the maximum weight that a 8-foot-long 2-by-4 piece of pine can support is 400 pounds. Write a general formula relating the maximum weight W (in pounds) to length l (in feet). Find the maximum weight W that can be safely supported by a length of 20 feet. Example : Modeling a Joint Variation: A Student’s GPA 1105 11e Notes Page 17 Example : Modeling a Joint Variation: A Student’s GPA A student’s GPA varies jointly with the average amount of time he or she studies each week and the average amount of time spent sleeping each week and varies inversely with the average amount of time spent on social media each week. Write an equation that relates these quantities. Example 4: Force of the Wind on a Window The force F of the wind on a flat surface positioned at a right angle to the direction of the wind varies jointly with the area A of the surface and the square of the speed v of the wind. A wind of 20 miles per hour blowing on a window measuring 3 feet by 4 feet has a force of 120 pounds. What force does a wind of 40 miles per hour exert on a window measuring 4 feet by 5 feet? 1105 11e Notes Page 18 4.1 Linear Function and Their Properties Average Rate of Change Linear Functions: f(x)=mx+b - The average rate of change (slope) is constant for linear functions. Example 1: Determine if each chart is linear or not. Time (x) Population (y) 0 0.09 1 0.12 2 0.16 3 0.22 4 0.29 5 0.39 Building Linear Models From Verbal Descriptions Example 3 1105 11e Notes Page 19 Example 4 Example 5 The simple cost function is the linear cost function, . Suppose that a small bike manufacturer has a daily cost of $1800 and each bike costs $90 to make. a) Write a linear model that expresses the cost C of manufacturing x bikes in a day. b) What is the cost to make 14 bikes in a day? c) How many bikes can be made for $3780? Example 6 Suppose the quantity supplied, S, and the quantity demanded, D, of a cell phone each month are given. S(p)=60p - 900 D(p)= -15p+2850 Where price p is the price in $. a) Find the equilibrium price (break-even) and quantity b) Determine the prices for which quantity supplied is greater than quantity demanded. 1105 11e Notes Page 20 1105 11e Notes Page 21 1105 11e Notes Page 22 4.3 Quadratic Functions and Their Properties Quadratic Function: Use the Vertex, Axis of Symmetry, and Intercepts to Graph a Quadratic Function. Example 1: Locate the vertex and axis of symmetry of the parabola defined by Example 2: For the quadratic , answer the following. a) Does the graph open up or down? b) What are the coordinates of the vertex? c) What is the equation of the axis of symmetry? d) What is/are the x-intercept(s)? e) What is the y-intercept? f) Graph the function. g) Determine the domain and range of the function. h) Determine where the function is increasing and decreasing. 1105 11e Notes Page 23 Does it open up or down? Find a Quadratic Function Given Its Vertex and a Point. 1. Plug in vertex as h and k (h, k) and plug in the given point x and y (x, y) to obtain a. 2. Plug a and the vertex in, then simplify. Example 3: Determine the quadratic function whose vertex is (1, -5) and whose y-intercept is -3. Example 4: Find the minimum or maximum of the given quadratic. Example 5 Determine the quadratic function whose vertex is (-2, -16) and whose x-intercepts is 2. *show MML graphing example Example 6 A water balloon is launched from a stadium 100 feet above the ground at an inclination of 45° to the horizontal, with an initial velocity of 100 feet per second. From physics, the height h of the projectile above the ground can be modeled by where x is the horizontal distance of the projectile from the base of the stadium. (a) Find the maximum height of the projectile. (b) How far from the base of the stadium will the water balloon hit the ground? 1105 11e Notes Page 24 4.4 Building Quadratic Models from Verbal description In economics, revenue R, in dollars, is defined as the amount of money received from the sale of an item and is equal to the unit selling price p, in dollars, of the item times the number x of units actually sold. That is, R = xp Example 1: Maximizing Revenue An electronics company has found that when certain calculators are sold at a price of p dollars per unit, the number x of calculators sold is given by the demand equation x = 18,750 – 125p. (a) Find a model that expresses the revenue R as a function of the price p. (b) What is the domain of R? (c) What unit price should be used to maximize revenue? (d) If this price is charged, what is the maximum revenue? (e) How many units are sold at this price? (f) Graph R. (g) What price should the company charge for the product to collect at least $590,625 in revenue? Example 2: Maximizing the Area Enclosed by a Fence 1105 11e Notes Page 25 Example 2: Maximizing the Area Enclosed by a Fence A farmer has 2500 yards of fence to enclose a rectangular field. What are the dimensions of the rectangle that encloses the most area? Example 3: The Golden Gate Bridge The Golden Gate Bridge, a suspension bridge, spans the entrance to San Francisco Bay. Its 746-foot-tall towers are 4200 feet apart. The bridge is suspended from two huge cables more than 3 feet in diameter; the 90-foot-wide roadway is 220 feet above the water. The cables are parabolic in shape and touch the road surface at the center of the bridge. Find the height of the cable above the road at a distance of 1500 feet from the center. 1105 11e Notes Page 26 3.1 Functions Relation: A relation is a correspondence between two sets: a set X, called the domain, and a set Y, called the range. In a re lation, each element from the domain corresponds to at least one element from the range. If x is an element of the domain and y is an element of the range, and if a relation exists from x to y, then we say that y c orresponds to x or that y depends on x and we write x → y It is often helpful to think of x as the input and y as the output of the relation. • • A relation can be expressed verbally. • A relation can be expressed numerically using a table of numbers or by using a set of ordered pairs. • A relation can be expressed graphically by plotting the points (x, y). • A relation can be represented as a mapping by drawing an arrow from an element in the domain to the corresponding element in the range. A relation is considered a function when no x-values repeat. Example 1: Describing a Relation A verbal description of a relation is given below. The average retail price of gasoline in the U.S. has changed over the years. In 2010, the average cost of one gallon of gasol ine was $2.78, in 2012 it cost $3.62, in 2014 it cost $3.36, in 2016 it cost $2.14, and in 2018 it cost $2.72. Using year as input and price as output, a) What is the domain and the range of the relation? b) Express the relation as a set of ordered pairs. c) Express the relation as a mapping. d) Express the relation as a graph. Example 2: Determining Whether a Relation Given by a Mapping is a Function For each relation, state the domain and range. Then determine whether the relation is a function. a) b) Example 3: Determining Whether a Relation Given by a Set of Ordered Pairs Is a Function For each relation, state the domain and range. Then determine whether the relation is a function. a) {(1, 5), (3, 9), (5, 1), (9, 2)} b) {(2, 6), (5, 5), (7, 5), (1, 15)} c) {(–5, 1), (–2, 2), (3, 3), (2, –2), (–5, –3)} 1105 11e Notes Page 27 Example 4: Determining Whether an Equation Is a Function Determine whether the equation y = 3x + 7 defines y as a function of x. Example 5: Determine whether the equation defines y as a function of x. 1105 11e Notes Page 28 Use Function Notation; Find the Value of a Function If f is a function, then for each number x in the domain, the corresponding number y in the range is designated by the symbol f(x), read as “f of x ” and we write y f x When a function is expressed in this way, we are using function notation. We refer to f (x) as the value of the function f at the number x. For example, the function y = 2x – 5 may be written using function notation as y = f(x) = 2x – 5. Example 6: Finding Values of a Function Important Facts about Functions • For each x in the domain of a function f, there is exactly one image f(x) in the range; however, more than one x in the domain can have the same image in the range. • f is the symbol that we use to denote the function. It is symbolic of the equation (rule) that we use to get from x in the domain to f(x) in the range. • If y = f(x), then x is the independent variable, or the argument of f, and y, or f(x), is the dependent variable, or the value of f at x. 1105 11e Notes Page 29 Difference Quotient Example 7: Finding the Difference Quotient Find the difference quotient of each function. a) b) Finding the Domain of a Function Defined by an Equation • If there is no fraction with a variable in the denominator or radical the domain is all real numbers. • If the equation has a fraction with a variable in the denominator, set the denominator equal to zero and solve. We do this b ecause we cannot have 0 in the denominator. • If the equation is just a radical or a fraction with a radical in the denominator, set the radicand (inside) greater than or equal to zero. We do this because we cannot have negative numbers inside the radical. • If you have an equation with a fraction having a radical on top and a variable in the denominator, set the radicand greater t han or equal to zero AND set the denominator equal to zero. Example 8: Find the Domain of a Function Example 2: Find the domain of each of the following: a) b) c) 1105 11e Notes Page 30 d) Form the Sum, Difference, Product, and Quotient of Two Functions Example 9: Operations on Function 1105 11e Notes Page 31 3.2 The graph of a Function Use the vertical line test. Example 1: Which graphs are graphs of a function? Obtain Information From a Graph. Example 2: d) Is f(-35) positive or negative? g) What is the range of f? e) For what value(s) of x is f(x)=0 h) What are the x-intercepts? i) What are the y-intercepts? f) For what values of x is f(x)>0? j) How often does the line y=1 intersect the graph? a) Find F(-35) and f(-15) b) Find f(30) and f(0) g) What is the domain of f? c) Is f (10) positive or negative? Example 3 Determine whether the graph is that is of a function by using the vertical line test. a) What is the domain and range? b) What are the intercepts? c) Find any symmetry. 1105 11e Notes Page 32 l) For what value(s) of x does f(x)=15? Example 4: a) b) c) d) e) f) Is the point (-1, 2) on the graph of F? If x=4, what is f(x)? What point is on the graph of F? If f(x)=2, what is x? What point(s) are on the graph of f? What is the domain of f? List the x-intercepts, if any? List the y-intercepts, if any? Example 5 A golf ball is hit with an initial velocity of 130ft/s at an inclination of 45 degrees to the horizontal. In physics, it is established that the height h of the golf ball is given by the function where x is the horizontal distance that the golf ball has traveled. a) Determine the height of the golf ball after it has traveled 100ft. b) What is the height after it has traveled 300ft? c) How far is the golf ball hit when it lands? 1105 11e Notes Page 33 3.3 Properties of Functions Determine Even and Odd Functions from a Graph Even: y-axis symmetry Odd: origin symmetry Example 1 Example 2: Identifying Even and Odd Functions from a Graph Determine whether each graph is the graph of an even function, an odd function, or a function that is neither even nor odd. Is f strictly decreasing on the interval (-2, 1)? Identify Even and Odd Functions from the Equation Example 2 1105 11e Notes Page 34 Use a Graph to Determine Where a Function is Increasing, Decreasing, or Constant Example 3 Determine where the function is increasing, decreasing, or constant. List the intercepts, if any, and find the domain and ra nge. Use a Graph to Locate Local Maxima and Local Minima Example 4 Use a Graph to Locate the Absolute Maximum and the Absolute Minimum 1105 11e Notes Page 35 Find the Average Rate of Change Example 5 Find the average rate of change of a) From 1 to 3 b) From 2 to 4 1105 11e Notes Page 36 1105 11e Notes Page 37 3.4 Library of Functions; Piecewise-Defined Functions Library of Functions 1105 11e Notes Page 38 Piecewise-Defined Functions Example 1 Example 2 Graph the piece-wise function: If a) b) c) d) Find the domain of the function. Locate and intercepts. Based on the graph, find the range. Is F continuous on its domain? 1105 11e Notes Page 39 Example 3 Graph, find the intercepts, domain, range. Example 4 A telephone company offers a monthly phone plan of $29.99. It includes 250 anytime minutes plus $0.30 per minute for additional minutes. The following function is used to compute the monthly cost for a subscriber, where x is the number of anytime minutes used. Compute the monthly cost of the cellular phone for the use of: a) 130min b) 315min 1105 11e Notes Page 40 3.5 Graphing Techniques: Transformations Example 1: Vertical Shift Up Use the graph of to obtain the graph of Find the domain and range of g. Example 2: Vertical Shift Down Use the graph of to obtain the graph of Find the domain and range of g. Vertical Shifts If a positive real number k is added to the output of a function y = f(x), the graph of the new function y = f(x) + k is the graph of f shifted vertically up k units. If a positive real number k is subtracted from the output of a function y = f(x), the graph of the new function y = f(x) – k is the graph of f shifted vertically down k units. Example 3: Horizontal Shift to the Right Use the graph of f(x) = |x| to obtain the graph of g(x) = |x – 2|. Find the domain and range of g. 1105 11e Notes Page 41 Example 4: Horizontal Shift to the Left Use the graph of f(x) = |x| to obtain the graph of g(x) = |x + 2|. Find the domain and range of g. Horizontal Shifts If the argument x of a function f is replaced by x – h, h > 0, the graph of the new function y = f(x – h) is the graph f shifted horizontally right h units. If the argument x of a function f is replaced by x + h, h > 0, the graph of the new function y = f(x + h) is the graph f shifted horizontally left h units. Example 5: Combining Vertical and Horizontal Shifts Graph the function . Find the domain and range of f. Example 6: Vertical Stretch Use the graph of to obtain the graph of 1105 11e Notes Page 42 Example 7: Vertical Compression Use the graph of to obtain the graph of of . Vertical Compression or Stretch If a function y = f(x) is multiplied by a positive number a, then the graph of the new function y = af(x) is obtained by multiplying each y-coordinate of the graph of y = f(x) by a. If 0 < a < 1, a vertical compression by a factor of a results. If a > 1, a vertical stretch by a factor of a results. Horizontal Compression or Stretch If the argument of a function y = f(x) is multiplied by a positive number a, then the graph of the new function y = f(ax) is obtained by multiplying each x-coordinate of the graph of y = f(x) by If a > 1, a horizontal compression by a factor of If 0 < a < 1, a horizontal stretch by a factor of results. results. Example 8: Reflection about the x-Axis Graph the function . Find the domain and range of f. Reflection about the x-Axis When a function f is multiplied by –1, the graph of the new function y = –f(x) is the reflection about the x-axis of the graph of the function f. 1105 11e Notes Page 43 Example 9: Reflection about the y-Axis Graph the function .Find the domain and range of f. Reflection about the y-Axis When the graph of the function f is known, the graph of the new function y = f(–x) is the reflection about the y-axis of the graph of the function f. Example 10 Find the function that is finally graphed after the following transformation are applied to a graph of y x in the order listed. (1) Shift up 10 units (2) Reflect about the x-axis (3) Reflect about the y-axis 1105 11e Notes Page 44 3.6 Mathematical Models: Building Functions Build and Analyze Functions Example 1: Example 2: 1105 11e Notes Page 45 Example 3: A rectangle has a corner in Quadrant 1 on the graph of the postive x-axis. a) Draw the figure. b) Express the area A of the rectangle as a function of x. c) What is the domain of A? 1105 11e Notes Page 46 another at the orgin, a third on the postive y-axis, and fourth on 6.1 Composite Functions Find a Composite Function and Domain To form a composite: - Put the second letter (that function) into the first letter (function) for every x and solve. Domain of a composite: - Find domain of the function you inputted (second letter). - Find the domain of the composite. Example 1 1105 11e Notes Page 47 Example 2 For a) b) Example 3 Find the composite functions and state the domain. a) b) 1105 11e Notes Page 48 6.2 One-to-One Functions; Inverse Functions Determine Whether a Function Is One-to-One A one-to-one function: Each x in the domain has one and only one image in the range. Meaning one x to one y and vice versa. Example 1: Determine which figure represents a one-to-one function Example 2 Determine the Inverse of a Function Defined by a Map or a Set of Ordered Pairs Example 3 1105 11e Notes Page 49 Example 4 Consider the functions a) Find f(g(x)). b) Find g(f(x)). c) Determine whether the functions f and g are inverses of each other. Find the Inverse of a Function Defined by an Equation Meaning: Switch x and y and solve for y. Example 5 Find the inverse if and check your answer. b) Find the domain and the range of f and 1105 11e Notes Page 50 Example 6 Example 7 The graph of a one-to-one function is given. Draw the graph of the inverse function. Example 8 The domain if a one-to-one function is [5, ∞), and its range is [-2, ∞), State the domain and range of its inverse function. 1105 11e Notes Page 51 1105 11e Notes Page 52 6.3 Exponential Functions Malthusian catastrophe as predicted by Malthus states that the growth of human population is governed by the rate of crop production. In the future, if the growth rate of crops decreases then human population growth is decreased. And there might come a day when we may have extremely limited supply of food which can cause widespread hunger and starvation. So exponential function is a reminder of how efficiently do we need utilize the available resources. "There is a continuous cycle of innovation that is necessary in order to sustain and avoid collapse. The catch, however, to this is that you have to innovate faster and faster. So the image is that we're not only on a treadmill that's going faster and faster, but we have to change the treadmill faster and faster. We have to accelerate on a continuous basis." - Geoffrey West Example 1 1105 11e Notes Page 53 Graph Exponential Functions Example 2 Example 3: Demo of MML Graphing Exponential Functions Using Transformations. Solve Exponential Equations Example 4 1105 11e Notes Page 54 Plot the point (0, 1), then move onto shifting and changing the base. Number e e It is approximately equal to 2.71828 Example 5 Solve: Example 6 Solve Example 7: Exponential Probability Between 9:00 pm and 10:00 pm, cars arrive at In-N-Out's drive-thru at the rate of 15 cars per hour (0.25 car per minute). The following formula from statistics can be used to determine the probability that a car will arrive within t minutes of 9:00 pm. F(t) = 1 – e–0.25t a) Determine the probability that a car will arrive within 5 minutes of 9 pm (that is, before 9:05 pm). b) Determine the probability that a car will arrive within 30 minutes of 9 pm (before 9:30 pm). c) Graph F. d) What does F approach as t increases without bound in the positive direction? 1105 11e Notes Page 55 1105 11e Notes Page 56 1105 11e Notes Page 57 1105 11e Notes Page 58 6.4 Logarithmic Functions Human senses, nearly all, work in a manner and obey Weber–Fetcher law, that response of the sense machinery is logarithm of an input. It is true at least for hearing, but also for eye sensitivity, temperature sense etc. And of course, in areas where it works normal ly. Because in extreme, there are other processes such as pain, etc. So as in a cause of hearing, what you experience is the logarithm of power of a sound wave, by "biological, natural, hear sen se construction. So, it is natural to use logarithmic units. Change Exponential Statements to Logarithmic Statements and Logarithmic Statements to Exponential Statements Example 1 Example 2 Example 3 Use a calculator to evaluate the expression. Round to three decimal places. 1105 11e Notes Page 59 Evaluating Logarithmic Functions Example 4 Determine the Domain of a Logarithmic Function The domain of a logarithmic function consists of the positive real numbers, so the argument of a logarithmic function must be greater than zero. Example 5 Solve Logarithmic Equations Example 6 1105 11e Notes Page 60 Example 7 Example 8 Solve the equation. Write in terms of the common logarithm. Example 9: Graphing a Logarithmic Function and Its Inverse 1105 11e Notes Page 61 Example 10: Graphing a Logarithmic Function and Its Inverse Example 11: Alcohol and Driving Blood alcohol concentration BAC is a measure of the amount of alcohol in a person’s bloodstream A BAC of % means that a person has 4 parts alcohol per 10,000 parts blood in the body. Relative risk is defined as the likelihood of one event occurring divided by the likelihood of a second event occurring. For example, if an individual with a BAC of 0.02% is 1.4 times as likely to have a car accident as an individual who has not been drinking, the relative risk of an accident with a BAC of 0.02% is 1.4. a) Research indicates that the relative risk of a person having an accident with a BAC of 0.02% is 1.4. Find the constant k i n the equation. b) Using this value of k, what is the relative risk if the concentration is 0.17%? c) Using this same value of k, what BAC corresponds to a relative risk of 100? d) If the law asserts that anyone with a relative risk of 4 or more should not have driving privileges, at what concentration of alcohol in the bloodstream should a driver be arrested and charged with DUI (driving under the influence)? 1105 11e Notes Page 62 6.5 Properties of Logarithms Work With Properties of Logarithms Example 1 Write each expression as a sum and/or difference of logarithms. Express powers as factors. a) log b) log c) ln d) ln e) ln 1105 11e Notes Page 63 Example 2 Write each of the expressions as a single logarithm. a) log log b) log log c) log log log Evaluate Logarithms Whose Base Is Neither 10 Nor e Example 3 Use the change-of-base formula and a calculator to evaluate the logarithm a) log b) log 1105 11e Notes Page 64 6.6 Logarithmic and Exponential Equations Example 1 Example 2 Example 3 Solve Exponential Equations 1105 11e Notes Page 65 Solve Exponential Equations Example 4 Example 5 Solve: a) b) Example 6 1105 11e Notes Page 66 Example 6 Solve: Example 7: Solving an Exponential Equation That Is Quadratic in Form Solve: 1105 11e Notes Page 67 6.7 Financial Models Continuous Compounding Theorem The amount A after t years due to principal P invested at an annual interest rate r compounded continuously is A = Pert Example 1: Using Continuous Compounding The amount A that results from investing a principal P of $2000 at an annual rate r of 8% compounded continuously for a time t of 1 year is: Effective Rate of Interest Theorem Example 2: Computing the Effective Rate of Interest-Which Is the Best Deal? Suppose you want to buy a 5-year certificate of deposit (CD). You visit three banks to determine their CD rates. American Express offers you 2.17% annual interest compounded monthly, and First Internet Bank offers you 2.18% compounded quarterly. Discover offers 2.16% compounded daily. Determine which bank is offering the best deal. The bank that offers the best deal is the one with the highest effective interest rate. Present Value Formulas 1105 11e Notes Page 68 Present Value Formulas Example 3: Computing the Value of a Zero-Coupon Bond A zero-coupon (noninterest-bearing) bond can be redeemed in 10 years for $1000. How much should you be willing to pay for it now if you want a return of 6% compounded continuously? Example 4: Determining the Time Required to Double or Triple an Investment a) How long will it take for an investment to double in value if it earns 6% compounded continuously? b) How long will it take to triple at this rate? 1105 11e Notes Page 69 Exponential Growth and Decay Models; Newton’s Law Law of Uninhibited Growth or Decay Many natural phenomena follow the law that an amount A varies with time t according to the function A(t) = A0ekt Here A is the original amount (t = 0) and k ≠ is a constant If k > 0, then equation (1) states that the amount A is increasing over time; if k < 0, the amount A is decreasing over time. In either case, when an amount A varies over time according to equation (1), it is said to follow the exponential law, or the law of uninhibited growth (k > 0) or decay (k < 0). Uninhibited Growth of Cells A model that gives the number N of cells in a culture after a time t has passed (in the early stages of growth) is N(t) = N0ekt k > 0, where N0 is the initial number of cells and k is a positive constant that represents the growth rate of the cells. Example 1: Bacterial Growth A colony of bacteria that grows according to the law of uninhibited growth is modeled by the function N(t) = 90e0.06t, where N is measured in grams and t is measured in days. a) Determine the initial amount of bacteria. b) What is the growth rate of the bacteria? c) What is the population after 7 days? d) How long will it take for the population to reach 250 grams? e) What is the doubling time for the population? Example 2: Bacterial Growth 1105 11e Notes Page 70 Example 2: Bacterial Growth A colony of bacteria increases according to the law of uninhibited growth. a) If N is the number of cells and t is the time in hours, express N as a function of t. b) If the number of bacteria doubles in 5 hours, find the function that gives the number of cells in the culture. c) How long will it take for the size of the colony to triple? d) How long will it take for the population to double a second time (that is, increase four times)? Uninhibited Radioactive Decay 1105 11e Notes Page 71 Uninhibited Radioactive Decay The amount A of a radioactive material present at time t is given by A(t) = A0ekt k < 0, where A0 is the original amount of radioactive material and k is a negative number that represents the rate of decay. Example 3: Estimating the Age of Ancient Tools Traces of burned wood along with ancient stone tools in an archeological dig were found to contain approximately 2.33% of the original amount of carbon-14. If the half-life of carbon-14 is 5730 years, approximately when was the tree cut and burned? Newton’s Law of Cooling The temperature u of a heated object at a given time t can be modeled by the following function: u(t) = T + (u0 – T)ekt k < 0 , where T is the constant temperature of the surrounding medium, u0 is the initial temperature of the heated object, and k is a negative constant. Example : Using Newton’s Law of Cooling An object is heated to 90°C (degrees Celsius) and is then allowed to cool in a room whose air temperature is 20°C. a) If the temperature of the object is 75°C after 5 minutes, when will its temperature be 50°C? b) Determine the elapsed time before the temperature of the object is 35°C. c) What do you notice about the temperature as time passes? 1105 11e Notes Page 72 5.1 Polynomial Functions Polynomial Function A polynomial function in one variable is a function of the form (standard form (highest exponent to lowest exponent)) where a , a … a , a are constants, called the coefficients of the polynomial, n ≥ 0 is an integer, and x is a variable. If a ≠ 0, it is called the leading coefficient, and n is the degree of the polynomial. The domain of a polynomial function is the set of all real numbers. Example 1: Identifying Polynomial Functions Determine which of the following are polynomial functions. For those that are, state the degree; for those that are not, state why not. Write each polynomial function in standard form, and then identify the leading term and the constant term. Summary of Properties If you take a course in calculus, you will learn that the graph of every polynomial function is both smooth and continuous. By smooth, we mean that the graph contains no sharp corners or cusps; by continuous, we mean that the graph has no gaps or holes and can be drawn without lifting your pencil from the paper. • • • • Real Zero If f is a function and r is a real number for which f(r) = 0, then r is called a real zero of f. As a consequence of this definition, the following statements are equivalent. r is a real zero of a polynomial function f. r is an x-intercept of the graph of f. x – r is a factor of f. r is a real solution to the equation f(x) = 0. 1105 11e Notes Page 73 • • • • r is a real zero of a polynomial function f. r is an x-intercept of the graph of f. x – r is a factor of f. r is a real solution to the equation f(x) = 0. Identifying the Real Zeros of a Polynomial Function and Their Multiplicity If a polynomial function f is factored completely, it is easy to locate the x-intercepts of the graph by solving the equation f(x) = 0 using the Zero-Product Property (setting the equation equal to zero). Example 2: Finding a Polynomial Function from Its Real Zeros Find a polynomial function of degree 3 whose real zeros are –1, 2, and 3. Example 3: Identifying Real Zeros and Their Multiplicities 1105 11e Notes Page 74 Example 4: Graphing a Polynomial Function Using Its x-intercept Consider the polynomial function: a) Find the x- and y-intercepts of the graph of f. b) Use the x-intercepts to find the intervals on which the graph of f is above the x-axis and the intervals on which the graph of f is below the x-axis. c) Locate other points on the graph, and connect the points with a smooth, continuous curve. Interval x y A/B? Turning Points If f is a polynomial function of degree n, then the graph of f has at most n – 1 turning points. (If given turning points the degree of the polynomial can be found by adding 1 to the number of turning points on the graph) Example 5: Identifying the Graph of a Polynomial Function Which graphs in the figures could be the graph of a polynomial function? For those that could, list the real zeros and state the least degree the polynomial can have. For those that could not, say why not. 1105 11e Notes Page 75 End Behavior of the Graph of a Polynomial Function The end behavior of the graph of the polynomial function is the same as that of the graph of the power function (y = LT) Example 6: Identifying the Graph of a Polynomial Function Example6: Which graph in the figures could be the graph of f (x) = x5 + ax3 + bx2 – 2x + 3 where a > 0, b > 0? Example 7: Finding a Polynomial Function from a Graph Find a polynomial function whose graph is shown (use the smallest degree possible). 1105 11e Notes Page 76 Find a polynomial function whose graph is shown (use the smallest degree possible). Example 8: Find the polynomial Given: Zeros: -2, multiplicity 2; 0; 5, multiplicity 2; Degree: 5 1105 11e Notes Page 77 5.2 Graphing Polynomial Functions; Models Example 1: Graphing a Polynomial Function Graph the polynomial function f(x) = (3x + 1)(x – 2)2. Example 2: Graphing a Polynomial Function Graph the polynomial function f(x) = (x + 1)(x + 3)2(x – 4). 1105 11e Notes Page 78 Example 3: Graphing a Polynomial Function Using a Graphing Utility Analyze the graph of the polynomial function: f(x) = x3 + 2.84x2 – 4.2251x + 2.34856 The figures shown are from a TI-84 Plus C. The y-intercept is f(0) = 2.34856. The graph of f Since it is not readily apparent how to factor f, use a graphing utility's ZERO (or ROOT or SOLVE) feature and determine the x-intercept is –4.03, rounded to two decimal places. The table shows values of x on each side of the x-intercept using a TI-84 Plus C. The points (–5, –30.53) and (–3, 13.58) rounded to two decimal places, are on the graph. From the graph of f, we see that f has two turning points. Using MAXIMUM reveals one turning point is at (–2.46, 15.04), rounded to two decimal places. 1105 11e Notes Page 79 Using MINIMUM shows that theBased other turning point fisisatincreasing (0.57, 1.05), rounded to two decimal on the graph, on the places. intervals (–∞ – and ∞ Also, f is decreasing on interval [–2.46, 0.57]. The figure shows the graph of f drawn by hand using the information in the previous steps 1105 11e Notes Page 80 5.3 Properties of Rational Functions Rational Function A rational function is a function of the form where p and q are polynomial functions and q is not the zero polynomial. The domain of R is the set of all real numbers, except those for which the denominator q is 0. Meaning: A rational function is when you have a fraction that has a polynomial on top and a polynomial on bottom. The domain is all real numbers, except the number that makes the bottome zero. Set the denominator equal to zero and solve. Exclude that value. Example 1: Finding the Domain of a Rational Function Horizontal and Vertical Asymptotes Let R denote a function. - If, as x → –∞ or as x → ∞ the values of R(x) approach some fixed number L, then the line y = L is a horizontal asymptote of the graph of R. - If, as x approaches some number c, the values |R(x → ∞ that is R(x → –∞ or R(x → ∞ then the line x = c is a vertical asymptote of the graph of R. Locating Vertical Asymptotes (VA) The graph of a rational function , 1105 11e Notes Page 81 The graph of a rational function , in lowest terms, has a vertical asymptote x = r if r is a real zero of the denominator q. That is, if x – r is a factor of the denominator q of the rational function R, in lowest terms, the graph of R has a vertical asymptote x = r. Meaning: The vertical asymptote is the domain. Example 2: Finding Vertical Asymptotes Finding Horizontal or an Oblique Asymptote of a Rational Function Example 3 1105 11e Notes Page 82 Example 3 Example 4 Example 5 Find the horizontal or oblique asymptote, if one exists, of the graph of: Example 6 Find the horizontal or oblique asymptote, if one exists, of the graph of: 1105 11e Notes Page 83 Find the horizontal or oblique asymptote, if one exists, of the graph of: Example 6 Example 7 1105 11e Notes Page 84 8.1/8.6/8.7 8.1 Systems of Equations 2 variable systems Solve Systems of Three Equations Containing Three Variables Example 1 Solve the system. If the system has no solution, say it is inconsistent. Example 2 1105 11e Notes Page 85 Example 3 Example 4 1105 11e Notes Page 86 8.6 Systems of Nonlinear Equations Example 1 Solve the system. Example 2 1105 11e Notes Page 87 Example 3 8.7 Systems of Inequalities Example 1 Graph the inequality 1105 11e Notes Page 88 Example 2 Graph *show graphing demo Example 3 A retired couple can invest up to $25,000. As their financial adviser, you recommend that they place at least $15,000 in Treasury bills yielding 2% and at most $5000 in corporate bonds yielding 3%. a) Using x to denote the amount of money invested in Treasury bills and y to denote the amount invested in corporate bonds, write a system of linear inequalities that describe the possible amounts of each investment. Assume that x and y are in thousands of dollars. b) Graph the system c) What are the corner points of the graph? 1105 11e Notes Page 89 1105 11e Notes Page 90