Y X Equations of Lines Y X

At the end of this lesson you will be able to: Write equations for non-vertical lines. Write equations for horizontal lines. Write equations for vertical lines. Use various forms of linear equations. Calculate the slope of a line passing through two points. Y X

Before we begin. Let’s review some vocabulary. Y X Slope (m) =  Y) Vertical change (  Y) Y-intercept (b): The y-coordinate of the point where the graph of a line crosses the y-axis. Slope (m): The measure of the steepness of a line; it is the ratio of vertical change (  Y) to horizontal change (  X).  X) Horizontal change (  X) X-intercept (a): The x-coordinate of the point where the graph of a line crosses the x-axis.

Equations of Non-vertical Lines. Let’s look at a line with a y-intercept of b, a slope m and let (x,y) be any point on the line. Y X Y-axis X-axis (0,b) (x,y)

Slope Intercept Form The equation for the non-vertical line is: Y X Y-axis X-axis (0,b) (x,y) YYYY XXXX y = mx + b y = mx + b ( Slope Intercept Form ) Where m is: m = YY XX = (y – b) (x – 0)

More Equations of Non-vertical Lines. Let’s look at a line passing through Point 1 (x 1,y 1 ) and Point 2 (x 2,y 2 ). Y X Y-axis X-axis (x 1,y 1 ) (x 2,y 2 )

Point Slope Form The equation for the non-vertical line is: Y X Y-axis X-axis YYYY XXXX y – y 1 = m(x – x 1 ) y – y 1 = m(x – x 1 ) ( Point Slope Form ) Where m is: m = YY XX = (y 2 – y 1 ) (x 2 – x 1 ) (x 1,y 1 ) (x 2,y 2 )

Equations of Horizontal Lines. Let’s look at a line with a y-intercept of b, a slope m = 0, and let (x,b) be any point on the Horizontal line. Y X Y-axis X-axis (0,b) (x,b)

Horizontal Line The equation for the horizontal line is still Y X Y-axis X-axis y = mx + b y = mx + b ( Slope Intercept Form ). Where m is: m = YY XX = (b – b) (x – 0)  Y = 0 XXXX (0,b) (x,b) = 0

Horizontal Line Because the value of m is 0, Y X y = mx + b becomes y = b (A Constant Function) Y-axis X-axis (0,b) (x,b)

Equations of Vertical Lines. Let’s look at a line with no y-intercept b, an x- intercept a, an undefined slope m, and let (a,y) be any point on the vertical line. Y X Y-axis X-axis (a,0) (a,y)

Vertical Line The equation for the vertical line is Y X Y-axis X-axis x = a x = a ( a is the X-Intercept of the line). Because m is: m = YY XX = (y – 0) (a – a) = Undefined (a,0) (a,y)

Vertical Line Because the value of m is undefined, caused by the division by zero, there is no slope m. Y X x = a becomes the equation x = a (The equation of a vertical line) Y-axis X-axis (a,0) (a,y)

Example 1: Slope Intercept Form Find the equation for the line with m = 2/3 and b = 3 Y X Y-axis X-axis Because b = 3  Y = 2  X = 3 (0,3)  X = 3 The line will pass through (0,3) Because m = 2/3 The Equation for the line is: y = 2/3 x + 3  Y = 2

Slope Intercept Form Practice Write the equation for the lines using Slope Intercept form. Y X 1.) m = 3 & b = 3 2.) m = 1 & b = -4 3.) m = -4 & b = 7 4.) m = 2 & b = 0 5.) m = 1/4 & b = -2

Example 2: Point Slope Form Let’s find the equation for the line passing through the points (3,-2) and (6,10) Y X Y-axis X-axis YYYY XXXX First, Calculate m : m = YY XX = (10 – -2) (6 – 3) (3,-2) (6,10) 3 12= =4

Example 2: Point Slope Form To find the equation for the line passing through the points (3,-2) and (6,10) Y X Y-axis X-axis YYYY XXXX y – y 1 = m(x – x 1 ) Next plug it into Point Slope From : (3,-2) (6,10) y – -2 = 4(x – 3) Select one point as P 1 : Let’s use (3,-2) The Equation becomes:

Example 2: Point Slope Form Simplify the equation / put it into Slope Intercept Form Y X Y-axis X-axis YYYY XXXX y + 2 = 4x – 12 Distribute on the right side and the equation becomes: (3,-2) (6,10) Subtract 2 from both sides gives. y + 2 = 4x – = - 2 y = 4x – 14

Point Slope Form Practice Find the equation for the lines passing through the following points using Point Slope form. Y X 1.) (3,2) & ( 8,-2) 2.) (-5,4) & ( 10,-12) 3.) (1,-5) & ( 7,7) 4.) (4,2) & ( -8,-4) 5.) (5,3) & ( 7,9)

Example 3: Horizontal Line Let’s find the equation for the line passing through the points (0,2) and (5,2) Y X Y-axis X-axis y = mx + b y = mx + b ( Slope Intercept Form ). Where m is: m = YY XX = (2 – 2) (5 – 0)  Y = 0 XXXX (0,2) (5,2) = 0

Example 3: Horizontal Line Because the value of m is 0, Y X y = 0x + 2 becomes y = 2 (A Constant Function) Y-axis X-axis (0,2) (5,2)

Horizontal Line Practice Find the equation for the lines passing through the following points. Y X 1.) (3,2) & ( 8,2) 2.) (-5,4) & ( 10,4) 3.) (1,-2) & ( 7,-2) 4.) (4,3) & ( -2,3)

Example 4: Vertical Line Let’s look at a line with no y- intercept b, an x-intercept a, passing through (3,0) and (3,7). Y X Y-axis X-axis (3,0) (3,7)

Example 4: Vertical Line The equation for the vertical line is: Y X Y-axis X-axis x = 3 x = 3 ( 3 is the X-Intercept of the line). Because m is: m = YY XX = (7 – 0) (3 – 3) = Undefined (3,0) (3,7) = 7 0

Vertical Line Practice Find the equation for the lines passing through the following points. Y X 1.) (3,5) & ( 3,-2) 2.) (-5,1) & ( -5,-1) 3.) (1,-6) & ( 1,8) 4.) (4,3) & ( 4,-4)

Equation Internet Activity Click on each of the links below and follow the directions to complete problems. SparkNotes: Slope Intercept Form SparkNotes: Point Slope Form Slope Intercept Form Information and Practice Point Slope Form Information and Practice

Graphing Equations Conclusions What are the similarities you see in the equations for Parallel lines? What are the similarities you see in the equations for Perpendicular lines? Record your observations on your sheet.

Equation Summary Slope: Slope (m) =  Y) Vertical change (  Y)  X) Horizontal change (  X) Slope-Intercept Form: y = mx + b Point-Slope Form: y – y 1 = m(x – x 1 )