<p>DAY 2 Slope Slope - think “ steepness, hills , roofs, steps”</p><p>A. How to find slope given a graph = Rise Run</p><p>• ( 2,3)</p><p>• (4, -1) Count from one dot to the next, going vertically then horizontally.</p><p>Rise = -4 Down four units = -2 * always reduce Run 2 Right two units 1</p><p>General type of slopes</p><p>Zero slope</p><p>Positive slope Negative slope Undefined slope</p><p>B. Finding slope between two points:</p><p>Slope = Y2 – Y1 Where ( x1, y1) and ( x2, y2) are two points X2 - X1</p><p>Ex # 1 Find the slope between the points ( 2, 3 ) and ( 7, 5 ) .</p><p>M = 5 – 3 = 2 7- 2 5 </p><p>Ex # 2 Find the slope between the points ( -4 , -3 ) and ( 6, -9 )</p><p>M= -9 - - 3 = - 9 + 3 = -6 = -3 * Always reduce slope to lowest terms 6 - - 4 6 + 4 10 5 DAY 3 Slope-intercept form</p><p>Slope intercept form: y = mx + b</p><p> m = slope and b = y-intercept</p><p> y-intercept is the point where the graph crosses the y axis.</p><p>Ex # 1 Carla has already written 10 pages of a novel. She plans to write 15 additional pages per month until she is finished.</p><p>A. Write an equation to find the total number of pages, P, written after any number of months. </p><p>15 pages per month is what is changing, so that is our slope.</p><p>10 pages already written is the starting point, so that is our y-intercept.</p><p>P = 15 m + 10</p><p>B. Graph the equation.</p><p>1. Put a dot on the starting point of 10. 2. Next pick a few values of m to plug in. P = 15(1) + 10 = 25 P = 15(2) +10 = 40 3. Plot these points and make a line. </p><p>C. Find the total number of pages written after 5 months.</p><p>Substitute 5 into the equation and solve for P.</p><p>P = 15(5) + 10 = 85 pages </p><p>Ex # 2 Write the equation of the line having a slope of 2 and a y-intercept of 5. Since m = 2 and b = 5 then y = 2x + 5 Ex # 3 Write the equation of the line shown in the graph.</p><p>Find slope. Count rise then run from one point to the other. m = 6/1 Find the y-intercept b = - 2</p><p> y = 6x - 2</p><p>Ex # 4 Graph the equation y = -2x + 3  Make sure your equation is solved for y before you begin.  Next find m and b. m = -2/ 1 b = 3  Then put a dot on the y-axis at b.  From the dot, count your rise then your run and put a second dot.  Connect the dots.</p><p> m = -2/ 1 b = 3 down 2 right 1</p><p>Ex # 5 Write a linear equation to model each situation.</p><p>An internet company charges $50 to install DSL plus $29.99 per month service fees.</p><p> m = 29.99 (per) b= 50 (starting point) so y = 29.99 x + 50.</p><p> Remember per and each represent what is changing, so they are clues for the slope. Day 4 Converting from standard form to slope-intercept form.</p><p>Slope- intercept form: y= mx + b</p><p>Standard form: Ax + By = C A, B and C are integers A is not negative</p><p> To convert between standard form and slope-intercept form, solve for y.</p><p>EX # 1 3x + 4y = 12 -3x -3x Subtract 3x from both sides 4y = -3x + 12 Write non-like terms next to each other. 4 4 4 Divide all 3 parts by the coefficient of y.</p><p> y = -3/4 x + 3</p><p>Ex # 2 2x – 6y + 9 =0 + 6y + 6y Since all terms are on the same side, move the y. 2x + 9 = 6y Divide all 3 parts by the coefficient of y. 6 6 6 </p><p>1/3 x + 3/2 = y Always reduce fractions when possible.</p><p>Horizontal and Vertical Lines </p><p>Since the line crosses the y axis at -3, the equation is y = -3 </p><p>Since the line crosses the x-axis at 2, the equation is x = 2 Ex # 3 a). Graph the equation X + 9 = 12 -9 -9 X = 3 </p><p> b). Graph the equation 2y + 7 = 0 -7 -7 2y = -7 2 2</p><p>Y = - 4 ½ Day 5 Writing Linear Equations</p><p>To write the equation of a line in slope-intercept form, you need the slope and the y-intercept.</p><p>EX # 1 Write the equation of a line with slope = 2 that passes through the point (3,4).</p><p>We know that m = 2 and that the ordered pair gives us an x and a y value. (x, y)</p><p> y = mx + b * Substitute for the variables you know.</p><p>4 = 2(3) + b * Solve for b 4 = 6 + b -6 -6 -2 = b * Use the b value and the m value to write a new equation.</p><p> y = 2x -2</p><p>EX # 2 Write the equation of the line that passes through the points (1, 3) and (-3,-5).</p><p>First find m using the formula: m = y2 – y1 = -5 – 3 = -8 = 2 = 2 x2 – x1 -3 – 1 -4 1 </p><p>Next pick one of the 2 ordered pairs to use for x and y. (1,3)</p><p>Substitute and solve for b. y = mx + b 3 = 2(1) + b 3 = 2 + b -2 -2 1 = b</p><p>Finally write a new equation y = 2x + 1 Day 6 Parallel and Perpendicular Lines</p><p>Parallel lines - Two lines in the same plane that never cross. - They have the SAME SLOPE.</p><p>Perpendicular lines - Two lines in the same plane that intersect at a right angle. ( 90 ◦) - Their slopes are negative reciprocals.</p><p>Ex. If m = 2 then perpendicular m = -1/2 Ex. If m = -3/4 then perpendicular m = 4/3</p><p>Think : “ flip and change sign”.</p><p>EX # 1 Write the equation of a line that passes through the point ( -2, 5) that is parallel to the line y = -4x + 2.</p><p> Since the lines are parallel, they must have the same slopes.  The slope of y = -4x + 2 is -4.  Use m = -4 and ( -2, 5) y = mx + b 5 = -4(-2) + b 5 = 8 + b -8 -8 -3 = b  Write the new equation y = - 4x – 3 </p><p>Ex # 2 Write the equation of the line passing through the point (3, -3) that is perpendicular to the line y = ¾ x + 5.</p><p>Since m = ¾ then perpendicular m = -4/3.</p><p> y = mx + b -3 = -4/3 (3) + b -3 = -4 + b +4 +4 1 = b Write the new equation y = -4/3 x + 1 </p>