Math 370 Precalculus [Note to Student: Read/Review Sec 1.1: The Distance and Midpoint Formulas] Sec 1.2: Graphs of Equations in Two Variables; Intercepts, Symmetry Defns A graph is said to be symmetric wrt (with respect to) the -axis, if for every point ( the point ( ) is also on the graph. (The graph is flipped across the -axis.) ) on the graph, A graph is said to be symmetric wrt the -axis if, for every point ( ) on the graph, the point ( ) is also on the graph. (The graph is flipped across the -axis.) A graph is said to be symmetric wrt the origin if, for every point ( ) on the graph, the point ( ) is also on the graph. (The graph is flipped across -axis followed by a flip across the -axis. It can also be viewed as a rotation.) Ex 1 The graph of an equation is given. a) Find the intercepts. b) Indicate whether the graph is symmetric wrt the -axis, the -axis, or the origin. -axis: -axis: : #39 #41 #46 Ex 2 List the intercepts and test for symmetry. a) (#56) b) (#57) Page 1 of 15
c) (#65) d) (#70) [Note to Student: Read/Review Sec 1.3: Lines] [Note to Student: We will come back to Sec 1.4: Circles] Interval Notation ( ) [ ] ( ] [ ) union (or) intersection (and) Ex 3 Solve, if necessary, each write answers using interval notation. Page 2 of 15
Sec 2.1: Functions Defns A relation is a correspondence between two sets. A relation can be thought of as a set of ordered pairs. If ( ) in an element of the relation, we say corresponds to or that depends on and we write. Let and be two nonempty sets. A function from to is a relation that associates with each element of exactly one element of. is called the domain of the function. For each element in, the corresponding element in is called the image of. The collection of the images is called the range of the function. ( is called the codomain of the function.) Ex 4 If ( ) is an ordered pair of the function, then. is an element of the whereas is an element of the. ( ) Notation Let. Consider What does ( ) ( ) mean? Page 3 of 15
Ex 5 Let be defined as ( ). Find the domain, range, and codomain of and illustrate using a diagram. Ex 6 Let ( ). Find ( ) ( ) and ( ). Ex 7 Find the domain of each function. a) (#47) b) (#49) c) (#51) ( ) ( ) ( ) d) (#55) e) (#58`) f) ( ) ( ) ( ) Need domain to be : ( ) ( ) Ex 8 Let ( ) ( ). Find ( ) ( ) and its domain. Page 4 of 15
Ex 9 (#75) Find the difference quotients for the function ( ). [Note to Student: This type of question on every exam.] ( ) ( ) ( ) ( ) Ex 10 Write as a composite of two functions. ( ) Sec 2.2: The Graph of a Function Vertical Line Test A set of points in the -plane is the graph of a function iff (if and only if) every vertical line intersects the graph in at most one point. Ex 11 notation. Determine if the graph is that of a function. Next, find the domain range. Write answers in interval Page 5 of 15
What does the HLT test for? Ex 12 (#9) Use the given graph of the function to answer pars (a)-(n). a) Find ( ) and ( ). b) Find ( ) and ( ). c) Is ( ) positive or negative? d) Is ( ) positive or negative? e) For what values of is ( )? f) For what values of is ( )? g) What is the domain of? h) What is the range of? i) What are the -intercepts? j) What is the -intercept? k) How often does the line intersect the graph? l) How often does the line intersect the graph? m) For what values of does ( )? n) For what values of does ( )? 11 Key Functions Refer to Key Function Handouts. Page 6 of 15
Sec 2.3: Properties of Functions Defns A function is even if, for every number in its domain, the number is also in the domain and ( ) ( ). A function is odd if, for every number in its domain, the number is also in the domain and ( ) ( ). Theorem A function is even iff its graph is symmetric wrt the -axis. A function is odd iff its graph is symmetric wrt the origin. Ex 13 Determine algebraically whether or not the function is even, odd, or neither. a) (#34) ( ) b) (#38) ( ) c) (#42) ( ) Ex 14 Find the slope of the given graph. Defn If and,, are in the domain of a function ( ), the average rate of change of from to is defined as ( ) ( ) Ex 15 (#56a) Find the average rate of change of ( ) from to. Page 7 of 15
Ex 16 The figure shows the velocity,, of a body moving along a coordinate line as a function of time,. Use the figure to answer each question. ( ) a) When is the object moving forward? Backwards? When does it change direction? ( ) b) When is the graph increasing? Decreasing? c) When is it accelerating? d) What is the body s acceleration at sec? e) What is the body s acceleration at sec? Sec 2.4: Library of Functions; Piecewise-defined Functions Ex 17 Let ( ) {. Find ( ) ( ) ( ) ( ) ( ). Ex 18 Write a function that describes the graph. Page 8 of 15
Ex 19 Do the following: (a) find the domain (b) locate any intercepts (c) graph the function (d) based on the graph, find the range. (#29) (#35) ( ) { ( ) { Sec 2.5: Graphing Techniques: Transformations Hw: Function Graphs Hw A & B Horizontal/VerticalShifting Vertically Stretched (narrower) vs Vertically Compressed (wider) Horizontally Stretched and Horizontally Compressed Reflection about the -axis and -axis Study Graphs of Functions: ( ) ( ) ( ) ( ) ( ) Ex 20 Match each graph with the following functions. Page 9 of 15
Ex 21 Write a function whose inverse is ( ) and has been transformed as specified: shifted right 6 units shifted down 3 units reflected across the -axis vertically stretched by a factor of 4 Ex 22 Graph ( ). Ex 23 ( ) Graph the power functions [Note to Student: Read/Review Sec 3.1: Linear Functions and Their Properties & Sec 3.2: Building Linear Functions from Data ] Page 10 of 15
Sec 3.3: Quadratic Functions and Their Properties Ex 24 (#29) Graph ( ) by first rewriting it in vertex form. Must CTS (complete the square). *Sec 2.6: Mathematical Models: Building Functions [*Graphing Technology Involved] Ex 25 (#2) Let ( ) be a point on the graph of. a) Express the distance from to the point ( ) as a function of. b) What is if? c) What is if? d) Use a graphing utility to graph ( ). e) For what value of is the smallest? Ex 26 (#9 ) A rectangle is inscribed in a circle of radius 2. See the figure. Let ( ) be the point in quadrant I that is a vertex of the rectangle and is on the circle. a) Express the area of the rectangle as a function of. b) Express the perimeter of the rectangle as a function of. Page 11 of 15
Ex 27 (#21) Inscribe a right circular cylinder of height and radius in a cone of fixed radius and fixed height. See the illustration. Express the volume of the cylinder as a function of. [Hint:. Note also the similar triangles.] Sec 3.4: Quadratic Models; Building Quadratic Functions from Data Defn In economics, revenue, in dollars, is defined as the amount of money received from the sale of an item and is equal to the unit selling price, in dollars, of the item times the number of units actually sold. That is,, where is number of units sold and is unit selling price in dollars Ex 28 (#5) The price (in dollars) and the quantity sold of a certain product obey the demand equation. a) Express the revenue as a function of. b) What is the revenue if 15 units are sold? c) What quantity maximizes revenue? What is the maximum revenue? d) What price should the company charge to maximize revenue? Page 12 of 15
Ex 29 (#9) 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 than can be enclosed? See the figure. Ex 30 (#19`) An accepted relationship between stopping distance, (in feet), and the speed of a car, (in mph), is on dry, level concrete. a) How many feet will it take a car traveling 45 mph to stop on dry, level concrete? b) If an accident occurs 200 feet ahead of you, what is the maximum speed you should be traveling to avoid being involved? Ex 31 Sec 3.5: Inequalities Involving Quadratic Functions Solve the inequality and write answer using interval notation. Page 13 of 15
Ex 32 Solve the inequality and write answer using interval notation. a) b) c) (#20) ( ) ( )( ) Ex 33 Math 400 Let ( ) ( ) and let. Write as a function of. Ex 34 Math 400 What does it mean for a function to be continuous at a point? What does it mean to be continuous? Find a value of to make continuous. ( ) { Page 14 of 15
Ex 35 Math 400 Determine when the Chain Rule applies. ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) Hint: Rewrite without exponent. ( ) ( ) ( ) ( ) ( ) Page 15 of 15