Chapter 3A -- Rectangular Coordinate System
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1 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3A! Page61 Chapter 3A -- Rectangular Coordinate System A is any set of ordered pairs of real numbers. A relation can be finite: {(-3, 1), (-3, -1), (0, 5), (1, -3), (2, 3)} This is a set of ordered pairs of real numbers, so it is a. Each ordered pair of real numbers corresponds to a on the Cartesian plane. The set of all points corresponding to a relation is the of the relation. Graph the relation given above. The of a relation is the set of all first elements of the ordered pairs. The of a relation is the set of all second elements of the ordered pairs. What is the domain of this relation?! What is the range of this relation?!
2 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3A! Page62 A relation can be infinite: Consider {(x, y) -3 x < 2, 1< y 4} List 3 different elements of that set:!!!! Graph the relation What is the domain of this relation? What is the range of this relation? Equations can be used to define relations Consider the set {(x, y) x + y = 3} List three elements of that set:!!!! It is important to note that the elements of this relation are the solutions to the equation. Graph the relation What is the domain of this relation? What is the range of this relation?
3 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3A! Page63 Consider the set {(x, y) x = y } 2 List several elements of that set. It is important to note that the elements of this relation are the solutions to the equation. What is the domain of this relation? What is the range of this relation? Consider the set {(x, y) y = x } 2 List several elements of that set. It is important to note that the elements of this relation are the solutions to the equation. What is the domain of this relation? What is the range of this relation?
4 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3A! Page64 Calculating Distance in the rectangular coordinate system: Find the distance d between the points A(1, 2) and B(-3, 4). Find the distance d between the points A (x 1, y 1 ) and B (x 2, y 2 ) Consider a general point B (x, y) that is 5 units from the origin. This equation is true for all points that are exactly 5 units from the origin. Sketch the graph of this equation.
5 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3A! Page65 Place the point B at (1, 2). List and graph 4 points that are 3 units from B. Is the origin 3 units from (1, 2)? Write an equation whose solution is all of the points that are 3 units from (1, 2) We could repeat this procedure starting at an arbitrary point (h, k) and find all the points that are a given distance called r from (h, k). In this case, we would find that This is the standard form of the equation of a centered at with radius. Write the equation of a circle centered at (-3, 0) with radius 2. What is the center of the circle (x + 4) 2 + (y 5) 2 = 6? What is the radius of this circle? Expand this equation and simplify:
6 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3A! Page66 The General Form of an Equation of a Circle is written Note that there is both an term and a term, but no term. The coefficients on the x 2 term and the y 2 are both Consider the equation 4x 2 + 4y 2 24x + 40y +135 = 0 Is this a circle? If so what is its center? What is its radius? Consider the equation 16x 2 +16y 2 +16x 56y + 5 = 0 Is this a circle? If so what is its center? What is its radius?
7 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3A! Page67 The Midpoint of a Line Segment. Consider the points A (-3, 2) and B ( 4, -4). Determine the midpoint of the segment connecting these 2 points. Remember: The number half between the numbers a and b is the average of a and b Given a line segment with endpoints (x 1, y 1 ) and (x 2, y 2 ), the midpoint M of that segment is Extra Problems: Text: 4, 5, 6a, 11-13, 14a, 15a, 16, 17a, 18-21
8 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3B! Page 68 Chapter 3B Graphs of Equations Graphs of twelve relations that you should know: y = c, c! Example : y = 3 Domain Range x = c, c! Example : x = 2 Domain Range y = x Domain Range y = x Domain Range
9 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3B! Page 69 y = x 2 y 2 = x Domain Range Domain Range y = x 3 y 3 = x Domain Range Domain Range
10 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3B! Page 70 x 2 + y 2 = c, c!, c > 0 Example x 2 + y 2 = 4 Domain Range y = c x 2.. c!, c > 0 Example:y = c x 2 Domain Range y = x xy = 1 Domain Range Domain Range
11 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3B! Page 71 Variations on the Basic Equations y = x! Domain! Range y = 2x 2! Domain! Range y = 2x 2 + 5! Domain! Range
12 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3B! Page 72 Intercepts The y-intercept is the point where the graph crosses the. It occurs when The x-intercept is the point where the graph crosses the. It occurs when Find the intercepts of the graph for the following equation: x 2 + xy + 5y 2 = 25. To find the x-intercept To find the y-intercept Find the intercepts of the graph for the following equation: (x 4) 2 + (y + 2) 2 = 9.
13 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3B! Page 73 Symmetry about the x-axis On the figure below, label the vertices in the first and second quadrants. Then reflect the image through the x-axis to create a polygon that is symmetric about the x-axis. Notice that if (a, b) is on the polygon, then so is A graph is symmetric about the x- axis iff for each (x, y) on the graph, is also on the graph. We can test an equation to see if its graph will be symmetric about the x-axis. Substitute -y for y in the equation. Simplify the equation. If the resulting equation is to the original equation, then the graph will be 1. Show that x + y 2 = 4 is symmetric about the x-axis.
14 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3B! Page 74 Symmetry about the y-axis On the figure below, label the vertices in the second and third quadrants. Then reflect the image through the y-axis to create a polygon that is symmetric about the y-axis. Notice that if (a, b) is on the polygon, then so is A graph is symmetric about the y- axis iff for each (x, y) on the graph, We can test an equation to see if its graph will be symmetric about the y-axis. Substitute -x for x in the equation. Simplify the equation. If the resulting equation is to the original equation, then the graph will be 2. Show that x 4 + y = 9 is symmetric about the y-axis.
15 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3B! Page 75 Symmetry about the origin Informally, a graph is symmetric about the origin if when it is rotated 180 about the origin, the graph looks the same. By definition, a graph is symmetric about the origin iff for each (x, y) on the graph, is also on the graph. Interestingly, if a graph is symmetric about the origin, then for each point on the graph, there is a corresponding point on the graph such that the line segment connecting these two points has the origin as its midpoint. We can test an equation to see if its graph will be symmetric about the origin. Substitute both -x for x and -y for y in the equation. Simplify the equation. If the resulting equation is to the original equation, then the graph will be 3. Show that x 2 = y is symmetric about the origin.
16 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3B! Page Determine the symmetry of y 3 = 6x 4 x a) Test for symmetry about the x-axis by substituting for Equivalent or Not Equivalent? Symmetric or Not Symmetric about x-axis? b) Test for symmetry about the y-axis by substituting for Equivalent or Not Equivalent? Symmetric or Not Symmetric about y-axis? c) Test for symmetry about the origin by substituting both! for and for Equivalent or Not Equivalent? Symmetric or Not Symmetric about origin? Suggested Problems:! Text: 8-14
17 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3C! Page 77 Chapter 3C -- Linear Equations in Two Variables Linear Equations Linear Equations Circle the equations below if it is an example of a linear equation in 2 variables. 3x + 4y = 5!! y = x 9 πx + 4y = 5 x + y + z = 0 3x + xy + 4y = 5 x + 4y = 2 x 2 + 4y = 2 y = 2 x v = 32t +16 y = 2 y = 2 x = 5 x = y Any equation that can be written in the form Ax + By + C = 0 where A and B are not both is called a
18 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3C! Page 78 You already know a lot about lines, but let s review quickly: Consider x + y = 4. x y When you create a table to list points to create the graph, you are determining the solutions to the equation. Solutions to linear equations in two variables are ordered pairs of real numbers that when substituted into the equation create an identity. The coordinates of all of the points on the line are solutions to the equation. How many solutions exist for this equation? Graphing lines with both intercepts Consider x + 3y = 3 When x = 0, y = and when y = 0, x = Consider 2x + y = 4. Solve this equation for y. What is the y -intercept?! When x = 0, y = What is the slope of this line?
19 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3C! Page 79 Slope Calculate the slope m of the line on the graph below.! m = Δy Δx =! Note Δ is the Greek letter.! Δ or D is for! Some people like the phrase!! slope = What is the equation of this line? This y = mx + b form of a linear equation is known as the form. The slope of a horizontal line is The slope of a vertical line is Parallel lines do not. Parallel lines have the same. When 2 lines are perpendicular their slopes are.
20 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 3C! Page 80 Determine the equation of the line through (-1, -2) that is perpendicular to y = 3x + 4. Determine the equation of the line through (-1, 0) and (4, 3). Consider a line segment through (-1, 0) and (4, 3). What are the characteristics of a perpendicular bisector of a line segment? Extra Problems: Text 17, 21a, 22 g, h, i, 23d, 27-29
21 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4A Page 81 Chapter 4A - Introduction to Functions Definition of a relation: A relation is Functions True or False:! A function is a relation in which no two different ordered pairs have the same first element. True or False:! A function is a relation where there is only one output (called the y-value), for each input (called the x-value). True or False:! A function is a rule that assigns exactly one element in a set B (called the range) to each element in a set A (called the domain). Which of the following represent a function? {(1, 2), (1, 3), (2, 3)} This set of ordered pairs is {(1, 3), (2, 3), (3, 3), (4, 3)} This set of ordered pairs is Functions as a Rule: 1. If f (x) = 3x + 4 then f (2) = 2. If f (x) = 3x 2 + 4x +1 then f ( 1) = Note: Order of operations dictates that 3 2 = while ( 3) 2 = These rules create sets of ordered pairs of real numbers ( x, f (x)) In these cases the domain is the set of numbers for which f (x) is defined. The range is the set of values that f (x) attains.
22 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4A Page 82 The set of all points corresponding to a function is the of the function. A curve in the xy plane is the graph of a function iff no line the curve So if there is a line that the curve then the curve is the graph of a This is called {(x, y) x = y } 2 Is a relation, but It the vertical line test. {(x, y) y = x } 2 Is a relation and a It the vertical line test. So what we are seeing is that there are 2 common ways to define a function. A function can be defined with an equation such as {(x, y) y = x } 2 or a function can be defined with a rule such as {(x, f (x)) f (x) = x } 2 But having seen all of this, the truth is that we seldom use the set notation, just the equation or the f (x) notation.
23 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4A Page 83 Difference Quotient. Assume that the function represented by this graph is a position function. The horizontal axis is time given in seconds. The vertical axis is distance an object has moved from the starting point given in feet. It will be helpful to remember that d = rt so r = What is the position of the object at t = 0? What is the position of the object at t = 4? How far did the object move during those 4 seconds? What is the average speed of the object during those 4 seconds? What is the position of the object at t = 8? How far did the object move during those 8 seconds? What is the average speed of the object during those 8 seconds? How far did the object move between t = 4 and t = 8 What is the average speed of the object during those 4 seconds? Connect the points on the graph for t = 4 and t = 8. What is the slope of that line segment? The slope of the secant line for a position function is How might you go about calculating an estimate for the instantaneous velocity at a point, say at t=4?
24 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4A Page 84 You might calculate the average velocity between t=4 and t=5. Or better, you might calculate the average velocity between t=4 and t=4.5. Or better yet, you might calculate the average velocity between t=4 and t=4.1 The smaller the interval of time, the closer the average velocity would be to instantaneous velocity. Here we have the graph of some function f (x) Choose a general point (x, f (x)) on the graph. Let h > 0 be some teeny-tiny number. x + h would be just to the right of x. (Imagine that we ve zoomed in on the graph so it looks big, but it s really small.) Plot the point at (x + h, f (x + h)). Connect those two points with a line segment. Calculate the slope of that line segment: This is what is called the. In calculus, you will study what happens as h gets smaller and smaller. In the meantime, we will practice the sometimes-messy algebra that surrounds the Difference Quotient. The difference quotient is a function of two variables, x and h.!!!! DQ(x, h) =
25 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4A Page If f (x) = x 2 +1, simplify the difference quotient: 2. If f (x) = x 1, simplify the difference quotient:
26 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4A Page If f (x) = x, simplify the difference quotient: Word Problems: Currents: 4. On a canoe trip, Amy and Brian paddled upstream (against the current) at a rate of 2 miles per hour relative to the riverbank. On the return trip downstream (with the current), their average speed was 3 miles per hour. Find Amy and Brian s paddling speed in still water and the speed of the river's current.
27 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4A Page 87 Mixtures: 5. Ten grams of sugar are added to a 40-g serving of a breakfast cereal that is 30% sugar. What is the percent concentration of sugar in the resulting mixture? 6. A lab assistant needed 20 ounces of a 10% solution of sulfuric acid. If she has 20 ounces of a 15% solution, how much must she draw off and carefully replace with distilled water in order to reduce it to a 10% solution?
28 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4A Page 88 Do a job problems I think of them this way: Amount of the job completed = (rate the job is completed) * (time working) Also remember that if it takes someone 3 hours to do a job, then the rate is 1 3 job per hour. 7. Amy can clean the house in 6 hours. Brian can do the same job in 9 hours. In how many hours can they do the job if they work together? 8. Amy and Brian can rake their entire yard in 2 hours when working together. If Brian requires 6 hours to do the job alone, how many hours does Amy need to do the job alone?
29 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4A Page One pipe can fill a tank 1.25 times faster than a second pipe. When valves to both pipes are opened, they fill the tank in five hours. How long would it take to fill the tank if only the second pipe is used? Write a function problems 10. Suppose there is a piece of sheet metal that is rectangular: 1 foot long and 2 feet wide. Four congruent squares are cut from the corners, so that the resulting piece of metal can be folded and welded into a box. If the length of the sides of the squares is s, write a function that describes the volume of the box in terms of s. 11. Suppose there is a wire 40 inches long. Suppose the wire is cut into 2 pieces, not necessarily equal in length. If each of the pieces of wire is bent to form a square, write a function that describes the sum of the areas of the two squares. Extra Problems: Text: 1, 2, 6, 7, 10 c-g, 11c-g, 12 c-g, 13, 14, 19-23
30 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4B Page 90 4B Graphs of Functions Graph the following library of basic functions. It is important to be able to recognize and sketch these graphs with ease! The constant function f (x) = c c is a real number Domain Range Identity function f (x) = x Domain Range Squaring Function f (x) = x 2 Cubing Function f (x) = x 3 Domain Range Domain Range
31 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4B Page 91 Square Root Function f (x) = x Domain Range Absolute Value Function f (x) = x Domain Range Reciprocal Function f (x) = 1 x Cube Root Function f (x) = 3 x Domain Range Domain Range =! 1 =! 0 =! = 8 =
32 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4B Page 92 Piecewise Functions x, x < 0 1. Let f (x) = 3, x = 0 x, x > 0 a) f ( 2) = b) f (4) = c) Graph f (x) d) x intercept e) y -intercept f) domain g) range Assume that the graph below is that of the function g(x). Evaluate g( 1) g 3 2 g( 3) Write a definition for g(x) Domain: Range:
33 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4B Page 93 Consider the graph on the left. Examine how the graph is changing as you look at it from left to right. On the left side, the graph is going down. That is the interval where the function is decreasing. At x = 1 2 there is a change. The graph starts to go up. This is the interval where the function is increasing. Increasing, Decreasing, and Constant A function f (x) is increasing on an interval I iff x 1 < x 2 x 1, x 2 I A function f (x) is decreasing on an interval I iff x 1 < x 2 x 1, x 2 I A function f (x) is constant on an interval I iff x 1 < x 2 x 1, x 2 I The graph below is associated with a function f (x) Domain Range x intercept y -intercept f (x) is increasing on f (x) is decreasing on f (x) is constant on Extra Problems:!! Text: 2-4, 6, 8
34 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4C Page 94 Chapter 4C -- Transformations of Functions Vertical Shifts x f (x) = x g(x) = x + 2 h(x) = x Domain of g(x) Domain of h(x) Range of g(x) Range of h(x) x intercept for g(x) x intercept for h(x) y -intercept for g(x) y -intercept for h(x) g(x) is increasing on h(x) is decreasing on In general, adding a constant to a function shifts the graph 1. Write a function g(x) that shifts the graph of f (x) = x 3 4 units down. 2. Write a function h(x) that shifts the graph of f (x) = x 3 units up.
35 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4C Page 95 Horizontal Shifts x f (x) = x g(x) = ( x 3) 2 h(x) = ( x + 2) 2 Domain of f (x), g(x), h(x) Range of f (x), g(x), h(x) vertex of g(x) vertex of h(x) x intercept for g(x) x intercept for h(x) y -intercept for g(x) y -intercept for h(x) g(x) is increasing on h(x) is decreasing on
36 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4C Page 96 x f (x) = x g(x) = x + 4 h(x) = x 1 For square root functions, there is a point where the x value makes the radicand equal 0. I call this the The anchor point for f (x) anchor point for g(x) anchor point for h(x) domain of g(x) domain of h(x) x intercept for g(x) x intercept for h(x) y -intercept for g(x) y -intercept for h(x) g(x) is increasing on h(x) is decreasing on Range of f (x), g(x), h(x)
37 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4C Page 97 In general adding a constant to x before applying the function results in a shift of the graph. 3. f (x) = x Write the function g(x) that shifts the graph of f (x) four units to the right. g(x) = Domain of g(x) The anchor point for g(x) 4. Write the function h(x) that shifts the graph of f (x) = x two units to the left. h(x) = vertex of h(x) = x intercept for h(x) y -intercept for h(x)
38 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4C Page 98 Reflections Let f (x) = x 2!!! Let g(x) = f (x) = x 2 Range of f (x)!!! Range of g(x) x f (x) = x 2 g(x) = x We can see f (x) is a of f (x) through the Evaluating f (-x) If f (x) = x 2 then f ( x) =!! If g(x) = 1 x then g( x) = If h(x) = x 3 then h( x) =!!
39 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4C Page 99 If r(x) = x then r( x)= Let p(x) = r( x) = What is the domain of p(x)? Graph r(x) = x and p(x) = x Consider f (x) = (x 3) 2 Let g(x) = f ( x) = Graph f (x) and g(x). Here we see that f ( x) is a of f (x) through the
40 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4C Page 100 Stretching and Shrinking x f (x) = x 3 g(x) = x
41 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4C Page 101 If c > 1, then the transformation y = cf (x) f (x) by a factor of c. Whereas if 0 < c < 1, then the transformation y = cf (x) f (x) by a factor of. Combinations of Transformations: 5. g(x) = 2 1 x 6 a) parent function b) domain of g(x) c) anchor point d) x intercept e) y -intercept!!! f) range g) g(x) is increasing on!! h) g(x) is decreasing on i) List the transformations in order because the order matters!
42 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4C Page 102 (x 2)2 6. h(x) = +1 4 a) parent function b) domain of h(x) c) vertex d) x intercept e) y -intercept!!! f) range g) h(x) is increasing on!! h) h(x) is decreasing on i) List the transformations in order because the order matters!
43 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4C Page 103 Even Functions: Notice if f (x) = x 2, then f ( x) = In other words f ( x) = A function is iff for every x in the domain of f (x) The graph of an even function has symmetry about the f (x) = x 2 g(x) = x 2 h(x) = x r(x) = 9 x 2 Odd Functions: Notice if f (x) = x 3, then f ( x) = In other words f ( x) = A function is iff for every x in the domain of f (x). The graph of an odd function has symmetry about the f (x) = x g(x) = x h(x) = x 3 r(x) = x 3
44 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4C Page 104 To determine if a function f (x) is even or odd or neither, simplify f ( x). If f ( x) = f (x), then f (x) is an function. If f ( x) = f (x), then f (x) is an function. 7. Determine if the following functions are even, odd, or neither a) f (x) = x 2 + 2x +1!!!!!!! b) g(x) = x 3 +1 c) h(x) = 3x 4 x 2 +1!!!!!!! d) q(x) = x2 +1 x 3 Extra Problems: Text: 1-12
45 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4D! 105 Chapter 4D - Maximum/Minimum Function Values Everything you wanted to know about Quadratic Functions 1) Let f (x) = (x +1) 2 4 Determine a) the vertex b) the axis of symmetry c) the minimum value of the function d) y-intercept e) Graph f (x) = (x +1) 2 4 2) Let g(x) = (x 2) 2 +1 Determine a) the vertex b) the axis of symmetry c) the maximum value of the function d) y-intercept e) Graph g(x) = (x 2) 2 +1
46 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4D! 106 It is nice when a quadratic function is given in standard form because it makes it easy to see that the vertex is at When a quadratic function is given in general form it is easy to determine that the y-intercept is But sometimes you will want to convert a quadratic function from general form into standard form by Converting quadratics from general form to standard form 3 Let f (x) = x 2 2x + 4 Complete the square a) vertex b) axis of symmetry c) minimum value of the function d) y-intercept e) Graph f (x) = x 2 2x + 4
47 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4D! Let h(x) = x 2 5x + 6 Complete the square a) vertex b) axis of symmetry c) minimum value of the function d) y-intercept e) Graph h(x) = x 2 5x Let f (x) = 3x 2 +12x + 4 Complete the square a) vertex b) axis of symmetry c) minimum value of the function d) y-intercept e) Graph f (x) = 3x 2 +12x + 4
48 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4D! Let g(x) = 4x 2 +12x +1 the square a) vertex b) axis of symmetry c) maximum value of the function d) y-intercept e) Graph g(x) = 4x 2 +12x Consider the function f (x) = ax 2 + bx + c a) vertex! b) axis of symmetry c) the extreme value of the function
49 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4D! 109 Definition: Extreme functional values are the maximum and/or minimum values of the function. Being able to determine the extreme value of a quadratic function is very important, but equally important is being able to determine where the graph of the function is intersecting the x-axis. This is called finding the roots or finding the zeros of the function because when the graph intersects the x-axis, the value of the function is zero. Finding the Zeros of Quadratic Functions To find the zeros of quadratic functions, we simply set the function equal to zero, and determine the values of x that make the equation true. This is called the equation. Finding the Zeros of Quadratic Functions by Completing the Square Look back at the graphs of these functions earlier in the section to see where the graphs intersect the x-axis. 8. f (x) = (x +1) 2 4!!!!!! 9. g(x) = (x 2) 2 +1 Notice that axis of symmetry is always half way between the zeros.
50 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4D! 110 Consider the quadratic function f (x) = x 2 + x 6. f ( 3) = and f (2) = and f (x) can be factored so that f (x) = This is not a coincidence! Consider the quadratic function f (x) = ax 2 + bx + c. f (m) = 0 and f (n) = 0 iff f (x) can be factored so that f (x) = a( x m) ( x n) In other words if a parabola crosses the x -axis at say x = 2 and x = 4, then the factors of the parabola s quadratic function are and If you know that a parabola intersects the x -axis at x = 3 and x = 2, can you be sure that its function is f (x) = (x + 3)(x 2)? In Chapter 2A, it was established that if ax 2 + bx + c = 0, then x = b 2 4ac is called It is often denoted by an uppercase D = Information about the roots of a quadratic can be learned by examining the
51 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4D! Given the parabola on the right, determine the quadratic function that yields this graph. Then find the determinant of that quadratic. When the discriminant is there are real roots. In these cases, the parabola intersects the x -axis.
52 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4D! Given the parabola on the left, determine the quadratic function that yields this graph. Then find the determinant of that quadratic. When the discriminant is there is real root. It is sometimes called a. In these cases, the parabola intersects the x -axis.
53 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4D! Given the parabola on the left, determine the quadratic function that yields this graph. Then find the determinant of that quadratic. When the discriminant is there are real roots. In these cases, the parabola intersect the x -axis.
54 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4D! 114 Applications 19. A toy rocket is launched from a 304 foot cliff. If the height, in feet, of the rocket is given by. f (t) = 16t t where t is the number of seconds after liftoff. Notice f (0) = This corresponds to the rocket starting What kind of function is f (t)? So the graph of this function is a The maximum value of this function will occur at the Rewrite f (t) in standard form: a) find the maximum height of the rocket. b) How long does it take for the rocket to reach that maximum height? c) When does the rocket hit the ground? d) What is the domain of f (t)?
55 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4D! Joe has 30 ft of fence to make a rectangular kennel for his dogs, but plans to use his garage as one side. What dimensions produce the greatest area? Definition: Extreme functional values are the maximum and/or minimum values of the function on its domain. Minimum value Minimum value Minimum value Maximum value Maximum value Maximum value
56 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4D! 116 Local Extrema There is a local maximum at x = a if there is an open interval I containing a, and x I It is also important to notice on the graph that at a local maximum, the graph changes from There is a local minimum at x = b if there is an open interval I containing b, and x I It is also important to notice on the graph that at a local minimum, the graph changes from 14) Consider the graph of the function f (x) = x 3 x 2 6x. a) What is the approximate value for the local maximum? b) About where does this local maximum occur? c) Find an approximate value for the local minimum. d) Indicate the approximate location of this minimum value. Note: Sometimes you will want to know the maximum (or minimum) value of the function.! Sometimes you will want to know where (for what x value) the maximum value occurs. Extra Problems: Text: 1, 2, 4, 6-32
57 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4E! 117 Chapter 4E - Combinations of Functions 1. Let f (x) = 4 x and g(x) = x + 5 The domain of f (x) is A = The domain of g(x) is B = a) ( f + g)(x) = For ( f + g)(x) to be defined, we will need AND so the domain of ( f + g)(x) is b) ( fg)(x) = For ( fg)(x) to be defined, we will need!!!!!!! so the domain of ( fg)(x) is c) f g (x) = The domain of f g (x) is Let f (x) be a function with domain A and let g(x) be a function with domain B. Then the domains of ( f + g)(x), ( f g)(x), and ( fg)(x) are alla B. the domain of f g (x) is { x A B, g(x) 0 }.
58 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4E! 118 Function Composition Definition For functions f (x) and g(x), the composition of functions ( f! g)(x) is defined as 2. Let f (x) = x + 5 and g(x) = 1 x + 3 a) ( f! g)( 2) = b) (g! f )( 2) = c) ( f! g)(x) = d) (g! f )(x) =
59 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4E! 119 Again, let f (x) = 4 x and g(x) = x + 5 then ( f! g) ( x) = To determine the domain of ( f! g) ( x) we would need So the domain of ( f! g) ( x) is The domain of ( f! g) ( x) is the set of all x in the domain of g(x) such that g(x) is in the domain of f (x). ( f! g) ( x) is defined whenever both g(x) and f (g(x)) are defined. 3. If ( f! g) ( x) = x x + 7, then f (x) = and g(x) =
60 Fry Texas A&M University! Fall 2016! Math 150 Notes! Section 4E! Here are the graphs of two functions. If the line is the graph of f (x) and the parabola is the graph of g(x), sketch ( f + g)(x) ( f + g)(x) = f ( 4) + g( 4) = f ( 2) + g( 2) = f (0) + g(0) = f (2) + g(2) = f (4) + g(4) = 5. Using the same graph and functions, determine a) ( f! g) ( 4) b) ( g! f )( 4) c) ( f! g) ( 0) d) ( g! f )( 0) Extra Problems: Text: 3, 8-11
61 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 4F! Page 121 Chapter 4F Inverse Functions Let f (x) = x and g(x) = x +1! Determine the following: a) ( f! g)(0)!!!!!!! b) (g! f )(0) c) ( f! g)(x)!!!!!! d) (g! f )(x) In basic terms two functions are said to be if. The notation for a function s inverse is It is important to understand that f 1 (x) does NOT represent In our example above, g(x) = f 1 (x) but we could also say One to one functions: Before we write a formal definition for an inverse function, we need to understand what it means for a function to be one-to-one. If f (x) is a function and if x 1 = x 2 can you be sure that f (x 1 ) = f (x 2 )? Why? For a function to have an inverse it has to be. Basically f (x) is a one-to-one function if no two elements in the domain correspond to the same element in the
62 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 4F! Page 122 The function f (x) = x 2 is not 1-1 because We see that there are two distinct elements in the domain that correspond to the By definition, f (x) is a one-to-one function if f (x 1 ) = f (x 2 ) implies that The function g(x) = x 3 is. If x 1 3 = x 2 3 then A function that is 1-1 will pass the That means that any line will intersect the graph of a one to one function at most. Notice how the graph of f (x) = x 2 the horizontal line test while the graph of g(x) = x 3. Definition: Let f (x) be a function with domain A and range B. Then the inverse function of f (x) is denoted has domain and range It is defined by f 1 (y) = x iff This means that if (a, b) is on the graph of f (x), then will be on the graph of f 1 (x). It is also important to note that f 1 ( f (x)) = for every x in and f ( f 1 (x)) = for every x in
63 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 4F! Page Consider f (x) = 1 x +1 a) What is the domain of f (x)? b) What is the range of f (x)? c) Show that f (x) a 1-1 function. To do this, we must show that Start by assume 1 x 1 +1 = 1 x 2 +1 Remember if two numbers are equal, their reciprocals are equal, so d) Find f 1 (x)! Step 1: Write the function with y instead of f (x) Step 2: Swap x and y Step 3: Solve for y.
64 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 4F! Page 124 So f 1 (x) = e) What is the domain of f 1 (x)? (Notice that the domain of f 1 (x) is the same as ) f) What is the range of f 1 (x)? (Notice that the range of f 1 (x) is the same as ) g) Prove that what we found for f 1 (x) really is the inverse of f (x). To do this we must verify that f 1 ( f (x)) = f ( f 1 (x)) = x h) Sketch both f (x) and f 1 (x) on the same graph at the top of page Consider f (x) = 4 2x!!! a) What is the domain of f (x)? b) What is the range of f (x)? c) Show that f (x) a 1-1 function: To do this, we must show that
65 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 4F! Page 125 Start by assuming that 4 2x 1 = 4 2x 2 Remember if two numbers are equal, their squares are equal, so d) Find f 1 (x) Step 1: Write the function with y instead of f (x) Step 2: Swap x and y Step 3: Solve for y. e) What is the domain of f 1 (x)? (Notice that the domain of f 1 (x) is the same as ) f) What is the range of f 1 (x)? (Notice that the domain of f 1 (x) is the same as ) h) Sketch both f (x) and f 1 (x) on the same graph
66 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 4F! Page 126 When a function is not 1-1, we sometimes restrict the of f (x) so that the function is then ) Consider f (x) = x 2. If we restrict the domain of f (x) to (, 0], then f (x) is 1-1 on this restricted domain a) On the restricted domain, what is the range of f (x)? b) Find f 1 (x) Step 1: Write the function with y Step 2: Swap x and y Step 3: Solve for y. c) What is the domain of f 1 (x)? d) What is the range of f 1 (x)? f e) Sketch both f (x) and f 1 (x) on the same graph.
67 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 4F! Page 127 It is important to note that the graph of f 1 (x) can be found by 4) Given that f (x) = 2x + 3 5x 7 is a one-to-one function on, ,, find its inverse function. Domain of f (x)! Range of f (x) Domain of f 1 (x)! Range of f 1 (x) Extra Problems:! Text: 3, 8-11
68 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 128 Chapter 5A (Graphing) Polynomials Polynomial Degree Leading Term Constant Term f (x) = 4x 3 + 2x 5 g(x) = 17x 5 + 6x 3 h(x) = πx 6 17 c(x) = 14 p(x) = p n x n + p n 1 x n 1 +!+ p 2 x 2 + p 1 x + p 0 where all of the exponents are non-negative integers, the p i 's are real numbers and p n 0 r(x) = 2 + 3x 5x 4 When we graph polynomials, we ll pay special attention to! the x and y intercepts! where the graph is above or below the x-axis! the behavior of the function as x approaches.! the behavior of the function as x approaches. We will use notation like this As x, x 3. It reads, As x goes to infinity, x cubed goes to infinity. It means that when the values of x are very large and positive, x cubed is very large and positive. In this case, the graph of the function goes up on the right side. As x, x 3. This reads, As x goes to negative infinity, x cubed goes to negative infinity. It means that when the values of x are very large and negative, x cubed is very large and negative. In this case, the graph of the function goes down on the left side. Later we will study functions that level off for large values of x and other functions that oscillate forever. In calculus you will learn a precise definition for very large.
69 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 129 Cubics: f (x) = x 3!!!!!!!! g(x) = (x 2) 3 Constant Term: Constant Term: y-intercept: y-intercept: Leading Term: Leading Term: As x, x 3 As x, (x 2) 3 As x, x 3 As x, (x 2) 3 x-intercept: x-intercept: Notice that the graph only intersects the x- axis in one place -- at x =. This is the only place that the function changes sign. When x < 0, x 3 < 0 When x > 0 x 3 > 0 Notice that the graph only intersects the x- axis in one place -- at x =. This is the only place that the function changes sign. When x < 2, (x 2) 3 > 0 When x > 2 (x 2) 3 < 0
70 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 130 h(x) = 2(x + 3)(x 1) 2 Constant Term: So the y-intercept is Leading Term: As x, 2(x + 3)(x 1) 2 As x, 2(x + 3)(x 1) 2 x-intercepts: Now we would like to solve 2(x + 3)(x 1) 2 > 0 (because when 2(x + 3)(x 1) 2 > 0, the graph of h(x) will be the x-axis.) Redraw the x-axis and plot the zeros list the factors to create a sign table determine the sign of each factor use the signs of the factors to determine the sign of the product h(x) > 0 when so that is where the graph of h(x) is above the x-axis
71 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 131 p(x) = 1 4 (x + 2)(x 2)(3x 4) Constant Term: So the y-intercept is Leading Term: As x, 1 4 (x + 2)(x 2)(3x 4) As x, 1 4 (x + 2)(x 2)(3x 4) x-intercepts: Solve p(x) > 0. Draw a number line and plot the zeros list the factors to create a sign table determine the sign of each factor use the signs of the factors to determine the sign of the product p(x) > 0 when so that is where the graph of p(x) is above the x-axis
72 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 132 True or False: Some cubics never intersect the x-axis. Some cubics intersect the x-axis in exactly one place. Some cubics intersect the x-axis in exactly two places. Some cubics intersect the x-axis in three places. Some cubics intersect the x-axis in four places. Quartics f (x) = x 4 +1!!!!!!! g(x) = 1 4 (x 2) 4 Constant Term: y-intercepts: x-intercepts: Leading Term: As x, x 4 +1 As x, x 4 +1 Constant Term: y-intercepts: x-intercepts: Leading Term: As x, 1 (x 2) 4 4 As x, 1 (x 2) 4 4
73 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 133 h(x) = x 3 (2x + 7) y-intercept Leading Term: As x, x 3 (2x + 7) As x, x 3 (2x + 7) x-intercepts: Solve h(x) = x 3 (2x + 7) > 0 p(x) = (x 2 1)(x + 2) 2 y-intercept Leading Term: As x, p(x) = (x 2 1)(x + 2) 2 As x, p(x) = (x 2 1)(x + 2) 2 x-intercepts: Solve p(x) = (x 2 1)(x + 2) 2 > 0
74 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 134 Polynomials of Higher Degree In general, to understand the behavior of a polynomial, 1. Plot the y-intercept 2. Determine the behavior of the polynomial for large positive values of x and for large negative values of x. The behavior of the polynomial at these extremes will be dominated by the leading term, the term with the highest power of x. 3. Find the x-intercepts 4. Determine where the polynomial is positive and negative because this will tell you where the graph is above and below the x-axis. f (x) = x 5!!!!!!! g(x) = x 5!!! Constant Term: Leading Term: Constant Term: Leading Term: As x, x 5 As x, x 5 As x, x 5 As x, x 5 x-intercepts: x-intercepts: x 5 > 0 x 5 > 0
75 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 135 h(x) = (3x 1) 2 (x 2)(x +1)(x + 3)!!!!!!! y-intercept is Leading Term: As x, h(x) = (3x 1) 2 (x 2)(x +1)(x + 3) As x, h(x) = (3x 1) 2 (x 2)(x +1)(x + 3) x-intercepts: Solve h(x) = (3x 1) 2 (x 2)(x +1)(x + 3) > 0
76 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 136 f (x) = x(x 3) 2 (2x + 5) 2 Degree of the polynomial: y-intercept: Leading Term: As x, x(x 3) 2 (2x + 5) 2 As x, x(x 3) 2 (2x + 5) 2 x-intercepts: Solve: x(x 3) 2 (2x + 5) 2 > 0
77 Fry Texas A&M University! Spring 2016! Math 150 Notes! Section 5A! Page 137 True or False: All linear functions intersect the x-axis. True or False: All polynomial functions of degree zero intersect the x-axis. True or False: All polynomial functions of degree one intersect the x-axis. True or False: All quadratic functions intersect the x-axis. True or False: All cubic functions intersect the x-axis. True or False: All quartic functions intersect the x-axis. True or False: All quintic functions intersect the x-axis. True or False: All polynomials functions of even degree intersect the x-axis. True or False: All polynomials functions of an odd degree intersect the x-axis. Extra Problems: Text:! 1-8
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