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Chapter 1 Linear Equations and Graphs Section 2 Graphs and Lines Part I Learning Objectives for Section 1.2 Graphs and Lines The student will be able to identify and work with the Cartesian coordinate system. The student will be able to draw graphs for equations of the form Ax + By = C. The student will be able to calculate the slope of a line. The student will be able to graph special forms of equations of lines. The student will be able to solve applications of linear equations. Barnett/Ziegler/Byleen Business Calculus 12e 2 The Cartesian Coordinate System The Cartesian coordinate system was named after René Descartes. It consists of two real number lines, the horizontal axis (x-axis) and the vertical axis (y-axis) which meet in a right angle at a point called the origin. The two number lines divide the plane into four areas called quadrants. The quadrants are numbered using Roman numerals as shown on the next slide. Each point in the plane corresponds to one and only one ordered pair of numbers (x,y). Two ordered pairs are shown. Barnett/Ziegler/Byleen Business Calculus 12e 3 The Cartesian Coordinate System (continued) II I (3,1) x III (–1,–1) Two points, (–1,–1) and (3,1), are plotted. Four quadrants are as labeled. IV y Barnett/Ziegler/Byleen Business Calculus 12e 4 Linear Equations in Two Variables A linear equation in two variables is an equation that can be written in the standard form Ax + By = C, where A, B, and C are constants (A and B not both 0), and x and y are variables. A solution of an equation in two variables is an ordered pair of real numbers that satisfy the equation. For example, (4,3) is a solution of 3x - 2y = 6. The solution set of an equation in two variables is the set of all solutions of the equation. The graph of an equation is the graph of its solution set. Barnett/Ziegler/Byleen Business Calculus 12e 5 Linear Equations in Two Variables (continued) If A is not equal to zero and B is not equal to zero, then Ax + By = C can be rewritten as 𝐴 𝐶 𝑦 = − x+ → 𝑦 = mx + b 𝐵 𝐵 This is known as slope-intercept form. If A = 0 and B is not equal to zero, then the graph is a horizontal line C y B If A is not equal to zero and B = 0, then the graph is a vertical line C x A Barnett/Ziegler/Byleen Business Calculus 12e 6 Using Intercepts to Graph a Line Graph 2x – 6y = 12. Barnett/Ziegler/Byleen Business Calculus 12e 7 Using Intercepts to Graph a Line Graph 2x – 6y = 12. x y 0 –2 y-intercept 6 0 x-intercept 3 –1 check point Barnett/Ziegler/Byleen Business Calculus 12e 8 Special Cases The graph of x = k is the graph of a vertical line k units from the y-axis. The graph of y = k is the graph of the horizontal line k units from the x-axis. Examples: 1. Graph x = –7 (this cannot be graphed on some calculators) 2. Graph y = 4 Barnett/Ziegler/Byleen Business Calculus 12e 9 Solutions x = –7 y=4 Barnett/Ziegler/Byleen Business Calculus 12e 10 Slope of a Line Slope of a line: y2 y1 rise m x2 x1 run x1 , y1 rise x2 , y2 run Barnett/Ziegler/Byleen Business Calculus 12e 11 Slope-Intercept Form The equation y = mx+b is called the slope-intercept form of an equation of a line. The letter m represents the slope and b represents the y-intercept. Barnett/Ziegler/Byleen Business Calculus 12e 12 Find the Slope and Intercept from the Equation of a Line Example: Find the slope and y intercept of the line whose equation is 5x – 2y = 10. Barnett/Ziegler/Byleen Business Calculus 12e 13 Find the Slope and Intercept from the Equation of a Line Example: Find the slope and y intercept of the line whose equation is 5x – 2y = 10. Solution: Solve the equation for y in terms of x. Identify the coefficient of x as the slope and the y intercept as the constant term. 5𝑥 − 2𝑦 = 10 −2𝑦 = −5𝑥 + 10 5 𝑦 = 𝑥−5 2 Therefore: the slope is 5/2 and the y intercept is –5. Barnett/Ziegler/Byleen Business Calculus 12e 14 Point-Slope Form The point-slope form of the equation of a line is y y1 m( x x1 ) where m is the slope and (x1, y1) is a given point. It is derived from the definition of the slope of a line: y2 y1 m x2 x1 Barnett/Ziegler/Byleen Business Calculus 12e Cross-multiply and substitute the more general x for x2 15 Example Find the equation of the line through the points (–5, 7) and (4, 16). Barnett/Ziegler/Byleen Business Calculus 12e 16 Example Find the equation of the line through the points (–5, 7) and (4, 16). Solution: 16 7 9 m 1 4 (5) 9 Now use the point-slope form with m = 1 and (x1, x2) = (4, 16). (We could just as well have used (–5, 7)). 𝑦 − 16 = 1(𝑥 − 4) 𝑦 = 𝑥 − 4 + 16 𝑦 = 𝑥 + 12 Barnett/Ziegler/Byleen Business Calculus 12e 17 Homework #1-2A: Pg 23 (3-51 mult of 3) Barnett/Ziegler/Byleen Business Calculus 12e 18 Chapter 1 Linear Equations and Graphs Section 2 Graphs and Lines Part II Using a Graphing Calculator Graph 2x – 6y = 12 on a graphing calculator and find the intercept. Barnett/Ziegler/Byleen Business Calculus 12e 20 Using a Graphing Calculator Graph 2x – 6y = 12 on a graphing calculator and find the intercept. Solution: First, we solve the equation for y. 2x – 6y = 12 Subtract 2x from each side. –6y = –2x + 12 Divide both sides by –6 y = (1/3)x – 2 Now we enter the right side of this equation in Y=, then Zoom Standard for the window dimensions, and graph the line. Barnett/Ziegler/Byleen Business Calculus 12e 21 Using a Graphing Calculator (continued) • To find the x-intercept • Go to CALC • Select 2: zero • Left bound? Position the cursor somewhere left of the x-intercept and hit ENTER • Right bound? Position the cursor somewhere right of the x-intercept and hit ENTER • Guess? Hit ENTER • X-intercept value appears at bottom of screen • To “plug in” a value for x • Go to TRACE • Enter x value (try 0), then ENTER • Y-coordinate appears at bottom of screen Barnett/Ziegler/Byleen Business Calculus 12e 22 Depreciation What is depreciation? Why do products depreciate? Give an example of a product that depreciates quickly. Give an example of a product that depreciates slowly. Barnett/Ziegler/Byleen Business Calculus 12e 23 Cost Analysis Office equipment was purchased brand new for $20,000 and will depreciate to a value of $2,000 after 10 years. 1. If its value depreciates linearly, find an equation that represents the depreciated value in dollars after t years. 2. What is the value of the equipment after 11 years? 3. When will the office equipment be worth half as much as it was brand new? Barnett/Ziegler/Byleen Business Calculus 12e 24 Cost Analysis (solution) 1. If its value depreciates linearly, find an equation that represents the depreciated value in dollars after t years. Answer: When t = 0, V = 20,000 and when t = 10, V = 2,000. Thus, we have two ordered pairs (0, $20000) and (10, $2000). Now, find the slope using the two points: 2000−20000 10−0 𝑚= = −1800 The y intercept is already known (when t = 0, V = 20000) b = 20000 Therefore, our equation is: V = –1,800t + 20,000. Barnett/Ziegler/Byleen Business Calculus 12e 25 Cost Analysis (solution) 2. What is the value of the office equipment after 11 years? • Set-up: t = 11 , V = ? V = -1800(11) + 20000 • Answer: V= $200 3. When will the office equipment be worth half as much as it was brand new? • Set-up: V = 10000, t = ? 10000 = -1800t + 20000 • Answer: t 5.6 years Barnett/Ziegler/Byleen Business Calculus 12e 26 Supply and Demand In a free competitive market, the price of a product is determined by the relationship between supply and demand. • If there is a surplus, the price tends to come down. • If there is a shortage, the price tends to go up. • The price tends to stabilize at the point of intersection of the demand and supply equations. This point of intersection is called the equilibrium point. The corresponding price is called the equilibrium price. The common value of supply and demand is called the equilibrium quantity. Barnett/Ziegler/Byleen Business Calculus 12e 27 What is a bushel? A bushel is a term used in agriculture. 8 quarts = 1 peck 4 pecks = 1 bushel ? Quarts = 1 bushel Barnett/Ziegler/Byleen Business Calculus 12e 28 Example At a price of $1.94 per bushel, the supply of corn is 9.8 billion bushels and the demand is 9.3 billion bushels. At a price of $1.82 per bushel, the supply of corn is 9.4 billion bushels and the demand is 9.5 billion bushels. 1. Find a price-supply equation of the form p = mx + b 2. Find a price-demand equation of the form p = mx + b 3. Find the equilibrium point algebraically. 4. Graph the equations on your graphing calculator and verify your answer in step 3. Barnett/Ziegler/Byleen Business Calculus 12e 29 Example (solution) 1. Find a price-supply equation of the form p = mx + b Supply data: 9.8 billion bushels $1.94 9.4 billion bushels $1.82 Ordered pairs: (9.8, $1.94) (9.4, $1.82) 1.82 − 1.94 𝑚= = 0.3 9.4 − 9.8 Use the point-slope form to find the equation of the line: 𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1 𝑦 − 1.94 = 0.3(𝑥 − 9.8) 𝑦 = 0.3𝑥 − 1 So, price-supply equation is: p = 0.3x – 1 Barnett/Ziegler/Byleen Business Calculus 12e 30 Example (solution) 2. Find a price-demand equation of the form p = mx + b Demand data: 9.3 billion bushels $1.94 9.5 billion bushels $1.82 Ordered pairs: (9.3, $1.94) (9.5, $1.82) 1.82 − 1.94 𝑚= = −0.6 9.5 − 9.3 Use the point-slope form to find the equation of the line: 𝑦 − 𝑦1 = 𝑚 𝑥 − 𝑥1 𝑦 − 1.94 = −0.6(𝑥 − 9.3) 𝑦 = −0.6𝑥 + 7.52 So, price-demand equation is: p = -0.6x+7.52 Barnett/Ziegler/Byleen Business Calculus 12e 31 Example (solution) 3. Find the equilibrium point algebraically. 0.3𝑥 − 1 = −0.6𝑥 + 7.52 0.9𝑥 = 8.52 𝑥 ≈ 9.47 𝑦 = 0.3𝑥 − 1 𝑦 = 0.3(9.47) − 1 𝑦 = 1.84 So the equilibrium point is (9.47, 1.84). This means the equilibrium price is $1.84 and the equilibrium quantity is 9.47 billion bushels. When the supply and demand are both 9.47 billion bushels, then the price per bushel is $1.84. Barnett/Ziegler/Byleen Business Calculus 12e 32 Example (solution) 4. Graph the equations on your graphing calculator and verify your answer in step 3. Barnett/Ziegler/Byleen Business Calculus 12e 33 Business – Markup Policy Urban Outfitters sells a shirt costing $20 for $33 and a jacket costing $60 for $93. A) If the markup policy of the store is assumed to be linear, write an equation that expresses retail price R in terms of wholesale cost C. B) What does the store pay for an outfit that retails for $240? Barnett/Ziegler/Byleen Business Calculus 12e 34 Business (solution) Urban Outfitters sells a shirt costing $20 for $33 and a jacket costing $60 for $93. Ordered pairs (wholesale, retail): (20, 33), (60, 93) 93 − 33 𝑚= = 1.5 60 − 20 𝑦 − 33 = 1.5(𝑥 − 20) 𝑦 = 1.5𝑥 + 3 𝑅 = 1.5𝐶 + 3 Barnett/Ziegler/Byleen Business Calculus 12e 35 Business (solution) What does the store pay for an outfit that retails for $240? 𝑅 = 1.5𝐶 + 3 𝑅 = $240 $240 = 1.5𝐶 + 3 C = $158 The store pays $158 for an outfit that retails for $240. Barnett/Ziegler/Byleen Business Calculus 12e 36 Homework #1-2B: Pg 24 (61-79 odd) Barnett/Ziegler/Byleen Business Calculus 12e 37