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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. 1 The Cartesian Coordinate System The Cartesian coordinate system was named after _____________________________________. It consists of two real number lines which meet in a right angle at a point called the ___________. The two number lines divide the plane into four areas called ______________________. 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. 2 The Cartesian Coordinate System (continued) II I Two points, (-1,-1) and (3,1), are plotted. (3,1) x III (-1,-1) The four quadrants are as labeled. IV y 3 Linear Equations in Two Variables A linear equation in two variables is an equation that can be written in the standard form _________________________, where A, B, and C are constants (A and B not both 0), and x and y are variables. A __________________ of an equation in two variables is an ordered pair of real numbers that satisfy the equation. For example, (4,3) is a ____________ of 3x - 2y = 6. The graph of the solution set (all of the ordered pair solutions) of a linear equation is a _____________. 4 Standard Form of a Linear Equation Ax + By = C If A is not equal to zero and B is not equal to zero, then A C Ax + By = C can be written as y x B B 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 5 Intercepts of a Line The INTERCEPTS of the graph of an equation are the points at which the graph intersects the horizontal (x) and/or vertical (y) axes. 6 Finding the Intercepts Algebraically 1. The x-intercept of an equation can be found algebraically by 7 Finding the Intercepts Algebraically 2. The y-intercept of an equation can be found algebraically by 8 Using Intercepts to Graph a Line Find the intercepts of the graph of the equation 2x - 6y = 12 algebraically, and use them to graph the line (on next slide). 9 Using Intercepts to Graph a Line Graph 2x - 6y = 12. x y 0 x-intercept 0 y-intercept 3 check point 10 Using a Graphing Calculator 1) Graph 2x - 6y = 12 on a graphing calculator and find the intercepts. (See handout) To Find Intercepts Using Graphing Calc 11 Other Examples 2) Graph 5x – 3y = 8 on a graphing calculator and find the intercepts. Write as ordered pairs. 3) Graph 3.6x – 2.1y = 22.68 on a graphing calculator and find the intercepts. Write as ordered pairs. 12 Special Case: Vertical Line The graph of x = k is the graph of a VERTICAL LINE that crosses the x-axis at (k, 0). Example: Graph x = -7 y x 13 Special Case: Horizontal Line The graph of y = k is the graph of the HORIZONTAL LINE that crosses the y-axis at (0, k). Example: Graph y = 3 y x 14 Writing Equations of Horizontal and Vertical Lines Example Graph the vertical and horizontal lines through the point (-5.5, 3), and then write the equations of each. 15 Horizontal and Vertical Lines Example Write each of the following equations in a simpler form by solving for the variable, and describe their graphs. 1. 5 y 13 21 2. 1 3 6 x0 2 4 16 Slope of a Line Given any two points in a plane, we can calculate the slope of the line through those points. Slope of a line: y2 y1 rise m x2 x1 run x1 , y1 rise x2 , y2 run 17 Slope of a Line Find the slope of the lines containing the following pairs of points: 1. 8, 7 and 2, 1 2. 2.8, 3.1 and 1.8, 2.6 18 Slope of a Line Find the slope of the lines containing the following pairs of points: 3. 5, 2 and 1, 2 4. 7, 4 and 7,10 19 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 _____________ and b represents the __________________________. 20 Important Note In general, you should express the slope as a FRACTION in simplest form. Always state the y-intercept as an ordered pair in the form (0, b). 21 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. Then graph this line. 22 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 Cross-multiply and substitute the more general x for x2 and y for y2 23 Example 1 Find the EQUATION of the line through the points (-5, 7) and (4, 16). Express your final answer in slope-intercept form. 24 Example 2 Find the EQUATION of the line through the points (2, 3) and (-3, 7). Express your final answer in slope-intercept form. 25 Example 3 Find the EQUATION of the line through the points (9, 0) and (9, -5). 26 Example 4 Find the EQUATION of the line through the points (-8, 1) and (-3, 1). 27 Name three points that are on the same VERTICAL line and then state the equation of the line. Name three points that are on the same HORIZONTAL line and then state the equation of the line. 28 Application Office equipment was purchased for $20,000 and will have a scrap value of $2,000 after 10 years. If its value is depreciated linearly, find the linear equation that relates value (V) in dollars to time (t) in years. 1. Since it is a linear function, we are looking for an equation in the form y = mx + b. Instead of x and y, our example uses the variables t and V. t is the ___________________________ variable. V is the __________________________ variable. So our answer will have the form of _______________________. 29 Application (continued) Office equipment was purchased for $20,000 and will have a scrap value of $2,000 after 10 years. If its value is depreciated linearly, find the linear equation that relates value (V) in dollars to time (t) in years. 2. Find the slope (m) of the equation (rate of change): m= 3. Do we know the y-intercept (b) (i.e. Do we know the value, V, when time, t, is 0?) 4. Substitute the values for m and b into the equation V = mt + b. 30 Additional Example from Text Page 27 # 62 31 Additional Example from Text Page 27 # 64 32