Finding Slopes and Equations of Lines in Point-Slope and Slope-Intercept Form, Exams of Calculus

Download Finding Slopes and Equations of Lines in Point-Slope and Slope-Intercept Form and more Exams Calculus in PDF only on Docsity! 1 MA 15200 Lesson 22 Sections 2.3 and 2.4 I Slope of a Line A measure of the ‘steepness’ of a line is called the slope of the line. Slope compares a vertical change (called the rise) to the horizontal change (called the run) when moving from one point to another point along a line. Slope is a ratio of vertical change to horizontal change. If a non-vertical line contains points ),( and ),( 2211 yxyx , the slope of the line is the ratio described by 2 1 2 1 change in y change in x y yrise m run x x − = = = − . *Note: Always be consistent in the order of the coordinates. There are 3 ways to find slope. 1. Using the slope formula (above) 2. Counting rise over run (when shown a graph) 3. Finding the equation in slope-intercept form (later in lesson) If a line is horizontal, the numerator in the slope formula will be 0 (the y coordinates of all points of a horizontal line are the same). The slope of a horizontal line is 0. If a line is vertical, the denominator in the slope formula will be 0 (the x coordinates of all points of a vertical line are the same). A number with a zero denominator is not defined or undefined. The slope of a vertical line is undefined. There are 4 types of slopes. Positive Negative Zero Undefined When given two points, it does not matter which one is called point 1 and which point 2. 2 1 1 2 2 1 1 2 y y y y x x x x − − = − − line rises left to right line falls left to right horizontal line vertical line Never say ‘no slope’ to define the slope of a vertical line. No slope could be interpreted as 0 or undefined. 2 Ex 1: Find the slope of a line containing each pair of points. Describe if the line rises from left to right, falls from left to right, is horizontal, or is vertical. ) (2, 3), ( 6, 12) ) ( 4, 2), (5,3) ) ( 1,0), (2,1) a P Q b P Q c P Q − − − − − ) ( 4,10), ( 4, 8) ) (6, 2), (9, 2) d P Q e P Q − − − − − II Equations of Lines, Point-Slope Form Begin with the slope formula and drop the subscript 2’s, putting them back as regular variables. 2 1 1 1 1 2 1 1 cross multiply ( ) y y y y m m y y m x x x x x x − − = → = → → − = − − − This is known as the point-slope form of the equation of a line. Point-Slope Form If a line contains the point 1 1( , ) and has the slope x y m , then the equation in point-slope form is 1 1( )y y m x x− = − . Ex 2: a) Write an equation in point-slope form for a line with a slope of 3 2 and through the point (2, 12). When using point-slope form, substitute values for 1 1, , and x y m . Never substitute for x and y. These are the variables of the equation. 5 Ex 8: Graph each line. 1 2 2 y x= + 3 4y x= − − V Equations and Graphs of Horizontal or Vertical Lines It a line is horizontal, the slope-intercept form is written 0 or y x b y b= + = . A vertical line cannot be written in slope-intercept form because there is no possible number for m. However, a vertical line would have points all with the same x-coordinate. So a vertical line can be written as x a= , where a is the x-intercept. If a and b are real numbers, then 2 2 x y 2 2 x y 6 • The graph of the equation x = a is a vertical line with an x-intercept of a. • The graph of the equation y = b is a horizontal line with a y-intercept of b. Note: If the equation has only an x or only a y, solve for that variable. Then you will know where the intercept is and be able to graph the line. Ex 4: Graph each line. ) 3 ) 4 a x b y = − = VI Intercepts In lesson 19 we discussed how to identify intercepts from a graph. How are intercepts found from an equation? It is easy to determine that at least one coordinate of an intercept is zero. An x-intercept will have the form ( ,0)a and a y-intercept will have the form (0, )b . Zero is your friend, when it comes to intercepts. Finding any intercepts: 1. Find the x-intercept by letting y = 0 and solving for x. 2. Find the y-intercept by letting x = 0 and solving for y. Ex 5: Find the x-intercept and y-intercept (if they exist) for these linear equations. 2 2 x y 7 ) 2 3 12 ) 5 12 9 ) 9 a x y b x y c x − = = − = Using Intercepts to Graph a Line 1. Plot the x-intercept. 2. Plot the y-intercept. 3. Draw a line through the two points that are the intercepts. Ex 6: Find the intercepts and use them to graph the line. 4 3 12 0x y− − = x y 2 2 x y