College Algebra Lecture Notes Section 2.2 Page 1 of 5 Section 2.2: Graphs of Linear Equations Big Idea: The graph of a line is always represented by the equation ax by c . Big Skill: .You should be able to graph a line given its equation, and determine when lines are parallel or perpendicular. A. The Graph of a Linear Equation Linear Equations A linear equation is one that can be written in the form ax by c where a and b are not both zero. Why can’t a and b both be zero? The graph of a linear equation is always a line. That’s why it’s called linear. See problem #1. Since there is only one line between any two points in a plane, you need only plot two points to graph any linear equation. Notice that most lines, like in problems#1 and #2, cross both the x-axis and the y-axis. The values at which the line crosses the axes are called the x-intercept and the y-intercept. Intercepts are convenient to use as the two points for graphing any line: The Intercept Method 1. Substitute x = 0 and solve for y. This gives the y-intercept (0, y). 2. Substitute y = 0 and solve for x. This gives the x-intercept (x, 0). 3. Plot the intercepts and draw a straight line between them. Practice: 1. Graph the equation 2 x 3 3 y 7 by plotting points. College Algebra Lecture Notes Section 2.2 Page 2 of 5 2. Graph the equation x y 2 by plotting two points. 3. Graph the equation 4 x 3 y 7 0 using the Intercept Method. B. The Slope of a Line The Slope Formula Given two points P1 x1 , y1 and P2 x2 , y2 , the slope of any nonvertical line through P1 and y2 y1 , where x1 x2. x2 x1 Be consistent when plugging in the x’s and y’s… Be careful to avoid sign errors when subtracting a negative. When m > 0, then the line between the points increases from left to right. When m < 0, then the line between the points decreases from left to right. The slope measures the rate of change of the y quantity with respect to the x quantity. P2 is: m College Algebra Lecture Notes Section 2.2 Page 3 of 5 Practice: 4. Compute the slope between the points (-3, -2) and (4, 6) and draw the line between them. Do the same for the points (-5, 5) and (8, -7). 5. A student lives 45 miles from MATC. He leaves home at 8AM and arrives at campus at 8:45 AM. Write an equation that models his distance from MATC as function of time and graph it. C. Horizontal and Vertical Lines Horizontal Lines The equation of a horizontal line is y = k, where (0, k) is the y-intercept. The Slope of a Horizontal Line The slope of any horizontal line is zero. Vertical Lines The equation of a vertical line is x = h, where (h, 0) is the x-intercept. The Slope of a Vertical Line The slope of any vertical line is undefined. College Algebra Lecture Notes Section 2.2 Page 4 of 5 Practice: 6. Graph x = 3 and y = -7 below. D. Parallel and Perpendicular Lines Parallel Lines Given that L1 and L2 are distinct non-vertical lines with slopes of m1 and m2 respectively, then: If m1 = m2, then the lines are parallel If L1 and L2 are parallel, then m1 = m2. Shortcut for the above two points: L1 and L2 are parallel if and only if m1 = m2. Shortcut for saying L1 and L2 are parallel: L1 || L2 Any two vertical lines are also parallel. Perpendicular Lines Given that L1 and L2 are distinct non-vertical lines with slopes of m1 and m2 respectively, then: If m1 m2, = -1 then the lines are perpendicular. If L1 and L2 are perpendicular, then m1 m2, = -1. Shortcut: L1 and L2 are perpendicular if and only if m1 m2, = -1. Shortcut for saying L1 and L2 are perpendicular: L1 L2 Any vertical line is perpendicular to any horizontal line. Practice: 7. Determine if the points (5, 1), (3, -2), and (-3, 2) form the vertices of a right triangle, and then find a pair of points parallel to one of the sides. College Algebra Lecture Notes D. Applications of Linear Equations Section 2.2 Page 5 of 5