Report

Linear Equations Slope-Intercept Form | Point-Slope Form | Horizontal & Vertical Lines Linear Equations Learning Objectives • Determine the equation of a line given the slope and y-intercept • Determine the equation of a line given the slope and any point • Determine the equation of a line given any two points on the line Slope-Intercept Form • Standard-form – an equation of a line in the form Ax + By = C – Value of A is greater than or equal to 0 – Value of A and B are not both 0 – Value of A, B, and C are real number constants • Ex) 3x + 2y = 6 – Subtract 3x from both sides of equation 2y = –3x + 6 – Then, divide both sides of equation by 2 for -3x y= +3 2 Slope-Intercept Form • Lines can be graphed by identifying the transformations on the linear parent function, y=x -3x – Ex) y = 2 + 3 • Reflected over the y-axis • Vertically stretched • Vertically translated up the y-axis by 3 • (0,3) is the y-intercept • Slope is –3/2 Slope-Intercept Form Slope-intercept form of a line The slope-intercept form of an equation of a line is in the form y = mx + b where m is the slope of the line, and b is the y-intercept. • Equation can be written quickly if slope and y-intercept are known • Slope and y-intercept recognized immediately in equation and line can be graphed quickly Point-Slope Form • When a slope is given without a y-intercept, an equation can be formed from a point on the line – Must apply transformations to the linear parent function until the equation represents the desired line • Ex) If slope is 1/2, then the graph of the linear parent function should be vertically compressed 1 y= x 2 • If the point on the line is (4,5), then translate the point to the right by four units and up by five units 1 y = ( x - 4) + 5 2 • Value of y for the given point can be moved to the other side of the equation 1 y - 5 = ( x - 4) 2 Point-Slope Form Point-slope form of a line The point-slope form of the equation of a line is y – y1 = m(x – x1) where m is the slope of the line, and (x1,y1) is a point on the line. • Advantages of point-slope form – Equation can be formed quickly from the slope and one point of a line – Can find y-intercept by solving the equation for y and converting the equation into slope-intercept form Point-Slope Form Example Ex) A line passes through the following two points, (1,4) and (5,7). Find the equation of this line in point–slope form and slope–intercept form. Analyze Formulate Justify Find slope and substitute into point–slope equation, solve for y Determine y1 - y2 7 - 4 3 = = x1 - x2 5 - 1 4 y= 3 ( x - 1) + 4 4 y - y1 = m( x - x1 ) y= 3 3 x- +4 4 4 3 y - 4 = ( x - 1) 4 y= 3 1 x+3 4 4 Slope is located to the left of the x-value in each form Evaluate Depending on the information given, certain forms can be created faster Horizontal & Vertical Lines • Sometimes linear equations do not appear to contain both x- and y-components – Ex) y = 4 • Two points on the line are (–3,4) and (5,4) • Slope is y1 - y2 4-4 0 = = =0 x1 - x2 5 - ( -3 ) 8 • Standard form is 0x + y = 4 • Slope–intercept form is y = 0x + 4 • Possible point–slope equation is y – 4 = 0(x – 5) Horizontal & Vertical Lines • Ex) x = 4 – Two points on the line are (4, –2) and (4,3) – Slope is y1 - y2 3 - ( -2) 5 = = = undefined x1 - x2 4-4 0 – Impossible to write equation of a vertical line in slope–intercept or point–slope form – Standard form is x + 0y = 4 Linear Equations Learning Objectives • Determine the equation of a line given the slope and y-intercept • Determine the equation of a line given the slope and any point • Determine the equation of a line given any two points on the line