3.7 Equations of Lines in the Coordinate Plane SOL G3a Objectives: TSW … investigating and calculating slopes of a line given two points on the line. write the equation of a line in slope-intercept form. write the equation of a line in point-slope form.

Slope (m) The ratio of its vertical rise to its horizontal run. Steepness Slope = m = Vertical rise Horizontal run

Example 1: Find the slopes

Slope (continued) of a line containing two points with coordinates (x 1, y 1 ) and (x 2, y 2 ) is given by the formula

Slopes All horizontal lines have a 0 slope All vertical lines have an undefined slope

Positive Slopes Rise (upward) as you move left to right Line slopes up from left to right y x

Negative Slope Fall (downward) as you move left to right Line slopes down from left to right y x

Example 2: Find the slope using the slope formula.

Rate of Change Describes how a quantity is changing over time. The slope of a line can be used to determine the Rate of Change Change in quantity (y) Change in time (x)

Example 3: Recreation: For one manufacturer of camping equipment, between 1990 and 2000 annual sales increased by $7.4 million per year. In 2000, the total sales were $85.9 million. If the sales increase at the same rate, what will be the total sales in 2010? mill. = y = y 2 – (10) = y 2 – 85.9

Forms of Linear Equations Slope-Intercept Form - y = mx + b slope y-intercept Point-Slope Form - y – y 1 = m(x – x 1 ) slope x-coordinate y-coordinate

Example 4: Graph 1.) The equation is in slope-intercept form y = mx + b The slope is y-intercept (0, 1) 2.) Plot the point (0, 1) 3.) Use the slope, from the point (0, 1) go up 2, right 3

Example 5: Graph 1.) The equation is in slope-intercept form y = mx + b The slope is 3 y-intercept (0, 1) 2.) Plot the point (0, 1) 3.) Use the slope 3, from the point (0, 1) go up 3, right 1

Example 6: Graph y - 3 = -2(x + 3) 1.) The equation is in point-slope form y – y 1 = m(x – x 1 ) The slope is -2 Point on line (-3, 3) 2.) Plot the point (-3, 3) 3.) Use the slope -2, from the point (-3, 3) go down 2, right 1

Example 7: Graph Point on line (4, 2) 2.) Plot the point (4, 2) The slope is 3.) Use the slope, from the point (4, 2) go down 1, right 3 1.) The equation is in point-slope form y – y 1 = m(x – x 1 )

Writing Equations of Linear Lines If we know the slope and at least one point If we have the slope and y-intercept, use the slope-intercept form; y = mx + b If we have the slope and a point, use the point-slope form; y – y 1 = m(x – x 1 )

Example 8: Write the equation of the line What is an equation of the line with slope 3 and y-intercept -5? Start with the slope-intercept form of the equation y = mx + b y = 3x + (-5) Substitute 3 for m, and -5 for b Simplify y = 3x - 5

Example 9: Write the equation of the line What is an equation of the line through point (-1, 5) with slope 2? Start with the point-slope form of the equation y – y 1 = m(x – x 1 ) y – 5 = 2(x - (-1)) Substitute 2 for m, and -1 in for x 1 and 5 in for y 1 Simplify y – 5 = 2(x + 1)

Example 10: Write the equation of the line What is an equation of the line with slope and y-intercept 2? Start with the slope-intercept form of the equation y = mx + b Substitute for m, and 2 for b y = x + 2

Example 11: Write the equation of the line What is an equation of the line through point (-1, 4) with slope -3? Start with the point-slope form of the equation y – y 1 = m(x – x 1 ) y – 4 = -3(x - (-1)) Substitute -3 for m, and -1 in for x 1 and 4 in for y 1 Simplify y – 4 = -3(x + 1)

Writing Equations of Linear Lines If we know two points on the line Find the slope using the formula Using the point-slope formula Plug in one of the two points Plug in the slope for m

Example 12: Write the equation of the line What is an equation of the line through point (-2, -1) and point (3, 5)? Find the slope y + 1 = (x + 2) ory - 3 = (x - 5) Start with the point-slope form of the equation y – y 1 = m(x – x 1 ) Plug in the slope and one of the two points

Writing Equations Horizontal and Vertical Lines We don’t need a slope All points on a horizontal line have the same y-coordinate; so the equation is y = y 1. All points on a vertical line have the same x-coordinate; so the equation is x = x 1. Where (x 1, y 1 )

Example 13: Write the equation of the line What are the equations for the horizontal and vertical lines through (2, 4)? The horizontal is y = y 1 y = 4 Substitute 4 for y 1 The vertical is x = x 1 x = 2 Substitute 2 for x 1

Example 14: Write the equation of the line What are the equations for the horizontal and vertical lines through (4, -3)? The horizontal is y = y 1 y = -3 Substitute -3 for y 1 The vertical is x = x 1 x = 4 Substitute 4 for x 1

Homework Pg 194 – 195 # 9 – 37 odds