David Raju 1.1 Lines

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. David Raju Y X

Let’s review some vocabulary. David Raju 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. David Raju Y X Y-axis X-axis (0,b) (x,y)

Slope Intercept Form The equation for the non-vertical line is: David Raju 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 ). David Raju 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: David Raju 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 =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. David Raju Y X Y-axis X-axis (0,b) (x,b)

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

Vertical Line The equation for the vertical line is David Raju Y X Y-axis X-axis x = a x = a ( a is the X-Intercept of the line). Because m is: m =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. David Raju 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 David Raju 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. David Raju 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) David Raju Y X Y-axis X-axis YYYY XXXX First, Calculate m : 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) David Raju 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 David Raju 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. David Raju 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) David Raju Y X Y-axis X-axis y = mx + b y = mx + b ( Slope Intercept Form ). Where m is: m =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, David Raju 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. David Raju 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). David Raju Y X Y-axis X-axis (3,0) (3,7)

Example 4: Vertical Line The equation for the vertical line is: David Raju Y X Y-axis X-axis x = 3 x = 3 ( 3 is the X-Intercept of the line). Because m is: m =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. David Raju Y X 1.) (3,5) & ( 3,-2) 2.) (-5,1) & ( -5,-1) 3.) (1,-6) & ( 1,8) 4.) (4,3) & ( 4,-4)

Graphing Calculator Activity Using a TI-84 calculator, graph the following equations. David Raju y 1 = 4x + 5 y 2 = ( 1/2 )x + 3 Y 3 = -2x + 2 y 4 = -(1/4)x + 1 y 5 = 4x + 0

Graphing Calculator Activity Describe the graphs of each of the lines. Include any similarities or differences you see in the graphs. Be sure to “Zoom Standard” and “Zoom Square” before you answer these questions. David Raju y 1 = 4x + 5 y 2 = ( 1/2 )x + 3 Y 3 = -2x + 2 y 4 = -(1/4)x + 1 y 5 = 4x + 0 Y-axis X-axis Press the space bar to compare your graphs with mine. The equation and it’s graph are color coded.

Graphing Calculator Activity Using a TI-84 calculator, graph the following equations. David Raju y 1 = 2x + 3 y 2 = ? Y 3 = -3x + -1 y 4 = ? y 5 = 7 y 6 = ? Now, graph each line given and a line that is Parallel to it on the calculator. Record the equations you use on your sheet.

Graphing Calculator Activity Compare the graphs of each set of lines. Be sure to “Zoom Standard” and “Zoom Square” before you compare graphs. David Raju Y-axis X-axis Press the space bar to compare your graphs with mine. The equations and their graphs are color coded. y 1 = 2x + 3 y 2 = ? Y 3 = -3x + -1 y 4 = ? y 5 = 7 y 6 = ?

Graphing Calculator Activity Using a TI-84 calculator, graph the following equations. David Raju y 1 = 2x + 3 y 2 = ? Y 3 = -3x + -1 y 4 = ? y 5 = 7 y 6 = ? Now, graph each line given and a line that is Perpendicular to it on the calculator. Record the equations you use on your sheet.

Graphing Calculator Activity Compare the graphs of each set of lines. Be sure to “Zoom Standard” and “Zoom Square” before you compare graphs. David Raju Y-axis X-axis Press the space bar to compare your graphs with mine. The equations and their graphs are color coded. y 1 = 2x + 3 y 2 = ? Y 3 = -3x + -1 y 4 = ? y 5 = 7 y 6 = ?

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. David Raju

Equation Summary David Raju 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 )