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Prepared by Doron Shahar Lines Prepared by Doron Shahar Warm-up: Class Notes Page 25 Today we are starting to learn about lines. In honor of that, here are several one-liners: I wondered why the Frisbee was getting bigger, and then it hit me. I had a dream I was eating a giant marshmallow, and when I woke up my pillow was missing! Right now I'm having amnesia and deja vu at the same time! I think I've forgotten this before? How’s that new line I introduced you to? Ohh, she goes on forever. Prepared by Doron Shahar Find points on a graph (−2,3) 4 x coordinate first y coordinate second 3 2 (2,1) (x , y) 1 0 -3 -2 -1-1 0 -2 -3 -4 1 2 3 4 5 6 Prepared by Doron Shahar Finding intercepts on a graph 4 3 2 y-intercept where line touches y-axis (0,2) 1 (4,0) 0 -3 -2 -1-1 0 -2 -3 -4 1 2 3 4 5 6 x-intercept where line touches x-axis Prepared by Doron Shahar Find slope from a graph 4 slope 3 rise run 2 2 1 Rise 0 -3 -2 -1-1 0 1 2 3 -2 -3 -4 Run 4 5 1 6 Prepared by Doron Shahar Slope formula The slope of a line passing through two distinct points m y 2 y1 x 2 x1 (x1, y1) and (x2, y2) is given by the formula____________ Exception: The slope is undefined if x1=x2. Extra 1: Find the slope of the line passing through the points (−2, −3) and (−4, 5). (x1, y1) and (x2, y2) m y 2 y1 x 2 x1 ( 5 ) ( 3 ) ( 4 ) ( 2 ) 8 2 4 Prepared by Doron Shahar Slope-intercept form The slope-intercept form of the equation of a line with slope m y mx b and y-intercept (0, b) is given by the formula _____________ 2.3.4 Find the slope-intercept form of the equation of the line with y-intercept (0, −8) and having slope m=2/5. y 2 5 x ( 8 ) Solution y 2 5 x8 Prepared by Doron Shahar Extra 2: Slope-intercept form Find the slope-intercept form of the equation of the line displayed in the graph below. 5 y-intercept (0, 3) (x1, y1) m 4 (2, 4) (x2, y2) 3 2 y 2 y1 x 2 x1 m 1 1 2 3 4 -2 -3 -4 (2) (0 ) 1 2 0 -5 -4 -3 -2 -1-1 0 ( 4 ) (3) Solution 5 y 1 2 x 3 Prepared by Doron Shahar 2.3.4 Sketch a graph Sketch a graph of the line y=2/5x+(−8). 3 2 y-intercept: (0, −8) 1 0 2 -1 0 1 2 3 4 5 6 7 8 9 10 -5 -4 -3 -2 -1 Slope: m -2 5 -3 -4 -5 -6 Two points define a line. -7 2 -8 -9 5 -10 Prepared by Doron Shahar 2.3.4 Identifying intercepts Find the y-intercept and the x-intercept of the line y=2/5x−8. To find the y-intercept, set x=0 and solve for y. Setting x equal to zero To find the x-intercept, set y=0 and solve for x. Setting y equal to zero y 2 /5 0 8 Solution y 8 y-intercept: ( 0 , 8 ) 0 2 / 5x 8 Solution x 20 x-intercept: ( 20 , 0 ) Prepared by Doron Shahar Point-slope form The point-slope form of the equation of a line passing through the point (x1, y1) and having slope m is given by the formula y y m ( x x ) 1 1 ______________________ Extra 3: Find the point-slope form of the equation of the line passing through the point (− 3,2) with slope m= −2. (x1, y1) y 2 2 ( x ( 3 ) ) Solution y 2 2 ( x 3) Prepared by Doron Shahar Extra 4: Point-Slope form Find the point-slope form of the equation of the line passing through the points (−1, 3) and (1, 4). (x1, y1) and (x2, y2) First find the slope. m y 2 y1 x 2 x1 y 3 1 2 ( x ( 1)) Solution y 3 1 2 ( x 1) ( 4 ) (3) (1) ( 1) 1 2 y ( 4 ) 1 2 ( x (1) ) Solution y 4 1 2 ( x 1) Prepared by Doron Shahar 2.3.2: Point-slope form Find the point-slope form of the equation of the line displayed in the graph below. 6 -6 (−4, 5) (x1, y1) -5 -4 -3 (−2, −3) (x2, y2) -2 5 4 3 2 1 0 -1-1 0 -2 -3 -4 -5 First find the slope. m 1 ( 3 ) ( 5 ) ( 2 ) ( 4 ) 8 2 4 2 Solution y 5 4 ( x 4) Prepared by Doron Shahar Extra 3: Sketch a graph Sketch a graph of the line y−2= −2(x+3). 5 4 1 Point Point: (−3, 2) 3 2 Slope: 21 m 2 0 -5 -4 -3 -2 -1-1 0 -2 -3 -4 -5 1 2 3 4 5 Two points define a line. Prepared by Doron Shahar 2.3.2 Identifying Intercepts Find the y-intercept and the x-intercept of the line y−5= −4(x+4). To find the y-intercept, set x=0 and solve for y. Setting x equal to zero y 5 4(0 4) Solution y 11 y-intercept: ( 0 , 11 ) To find the x-intercept, set y=0 and solve for x. Setting y equal to zero 0 5 4( x 4) Solution x 11 4 x-intercept: ( 11 4 , 0 ) Prepared by Doron Shahar Point-slope to Slope-intercept form 2.3.2 Write the line y−5=−4(x+4) in slope-intercept form. Starting Equation y 5 4( x 4) Distribute right hand side y 5 4 x 16 Slope-intercept form 5 5 y 4 x 11 Prepared by Doron Shahar Standard form of a line The standard from of the equation of a line is given by the formula Ax By C __________________ , where A and B are not both zero. 2.3.1 Write x/3 − 4y +1 =0 in standard form such that A, B, C are integers. Subtract 1 from both sides Multiply both sides by 3 x 4 y 1 3 x 12 y 3 Prepared by Doron Shahar 2.3.1 Identifying Intercepts Find the y-intercept and the x-intercept of the line x−12y= −3. To find the y-intercept, set x=0 and solve for y. Setting x equal to zero 0 12 y 3 Solution y 1/ 4 y-intercept: To find the x-intercept, set y=0 and solve for x. Setting y equal to zero x 12 0 3 Solution x 3 ( 0 , 1 4 ) x-intercept: ( 3 , 0 ) Prepared by Doron Shahar 2.3.1 Sketch a graph Sketch a graph of the line x−12y= −3. 2 x-intercept: (−3, 0) 1.5 x-intercept y-intercept: (0, 1/4) y-intercept 1 0.5 0 -5 -4 -3 -2 -1 -0.5 -1 -1.5 -2 0 1 2 3 4 5 Two points define a line. Prepared by Doron Shahar Standard form Every line can be written in standard form. This is not true of the other two forms. In particular, verticals lines can only be written in standard form, because their slope is undefined. It is difficult to get the equation of a line directly into standard form given a graph or “usual” information. Instead, you will need to be able to get the equations of lines into and out of this form. Prepared by Doron Shahar Point-slope to Standard form To get the equation of a line from point-slope form into standard form, first convert it into slope-intercept form as we have already learned. Then proceed as on the next slide to convert the equation into standard form. Prepared by Doron Shahar Slope-intercept to Standard form 2.3.4 Write the line y=2/5x−8 in standard form such that A, B, and C are integers. y 2 5x8 Starting Equation 2 5x 2 5x 2 5 x y 8 Multiply both sides by 5 5 2 5 x y 5 8 Distribute the 5 Alternative Standard form 2 x 5 y 40 2 x 5 y 40 Prepared by Doron Shahar Standard to Slope-intercept to form 2.3.1 Write the line x−12y = −3 in slope-intercept form. Starting Equation Subtract x from both sides Divide by −12 Slope-intercept form x 12 y 3 x x 12 y x 3 y y x3 12 1 12 x 1 4 Prepared by Doron Shahar Standard to Slope-intercept form One reason for converting the equation of a line from Standard form to slope-intercept form is to find the slope of the line. It also helps if you are trying to graph the line. 2.3.1 Find the slope of the x−12y = −3. Slope-intercept form y 1 12 Slope: m 1 12 x 1 4 Prepared by Doron Shahar Turtle Sign language for turtle Prepared by Doron Shahar Horizontal lines A horizontal line passing through the point (a, b) is given y b by the equation ________. m 0 The slope of a horizontal line is _________. 2.3.5 Write the equation of the horizontal line passing through the point (−9/2, 15/2). (a , b) Solution y 15 2 Prepared by Doron Shahar Extra 5: Horizontal lines Find the equation of the horizontal line displayed below. 5 4 (2, 3) (a, b) 3 2 Solution 1 0 -5 -4 -3 -2 -1-1 0 -2 -3 -4 -5 1 2 3 4 5 y 3 Prepared by Doron Shahar 2.3.5 Sketch a graph Sketch a graph of the line y=15/2. (−9/2, 15/2) 9 Slope: 8 7 6 5 4 3 2 1 0 -5 -4 -3 -2 -1-1 0 -2 1 2 3 4 5 m 0 Prepared by Doron Shahar 2.3.5 Identifying intercepts Find the y-intercept and the x-intercept of the line y=15/2. To find the y-intercept, set x=0 and solve for y. Setting x equal to zero y 15 / 2 Solution y-intercept: y 15 2 To find the x-intercept, set y=0 and solve for x. Setting y equal to zero 0 15 / 2 ¡PROBLEMA! 0 15 / 2 ( 0 , 15 2 ) There is no x-intercept! Prepared by Doron Shahar Vertical lines A vertical line passing through the point (a, b) is given xa by the equation ________. undefined The slope of a vertical line is _____________. 2.3.3 Write the equation of the vertical line with x-intercept (1/3, 0). (a , b) Solution x 1 3 Prepared by Doron Shahar Extra 6: Vertical lines Find the equation of the vertical line displayed below. 5 4 (2, 3) (a, b) 3 2 Solution 1 0 -5 -4 -3 -2 -1-1 0 -2 -3 -4 -5 1 2 3 4 5 x2 Prepared by Doron Shahar 2.3.3 Sketch a graph Sketch a graph of the line x=1/3. 5 Slope: undefined 4 3 2 1 (1/3, 0) 0 -2 -1 -1 0 -2 -3 -4 -5 1 2 Prepared by Doron Shahar 2.3.3 Identifying intercepts Find the y-intercept and the x-intercept of the line x=1/3. To find the y-intercept, set x=0 and solve for y. Setting x equal to zero To find the x-intercept, set y=0 and solve for x. Setting y equal to zero x 1/ 3 0 1/3 ¡PROBLEMA! 0 1 / 3 There is no y-intercept! Solution x-intercept: x 1/ 3 (1 / 3 , 0 ) Prepared by Doron Shahar Horizontal lines vs. Vertical lines Horizontal Lines Verticals lines Equation yb xa Slope m 0 undefined x-intercept y-intercept none Exception: y=0 (a, 0 ) ( 0 ,b ) none Exception: y=0