Graphing - Point-Slope Form

Transcription

1 . Graphing - Point-Slope Form Objective: Give the equation of a line with a known slope and point. The slope-intercept form has the advantage of being simple to remember and use, however, it has one major disadvantage: we must know the y-intercept in order to use it! Generally we do not know the y-intercept, we only know one or more points (that are not the y-intercept). In these cases we can t use the slope intercept equation, so we will use a different more flexible formula. If we let the slope of an equation be m, and a specific point on the line be (x 1, y 1 ), and any other point on the line be (x, y). We can use the slope formula to make a second equation. Example 1. m, (x 1, y 1 ), (x, y) Recall slope formula y y 1 =m Plug in values x x 1 y y 1 =m Multiply both sides by (x x 1 ) x x 1 y y 1 =m(x x 1 ) If we know the slope, m of an equation and any point on the line (x 1, y 1 ) we can easily plug these values into the equation above which will be called the pointslope formula. Point Slope Formula: y y 1 = m(x x 1 ) Example. Write the equation of the line through the point (, ) with a slope of. y y 1 = m(x x 1 ) y ( )= (x ) y + = (x ) Plug values into point slope formula Often, we will prefer final answers be written in slope intercept form. If the directions ask for the answer in slope-intercept form we will simply distribute the 1

2 slope, then solve for y. Example. Write the equation of the line through the point ( 6, ) with a slope of in slope-intercept form. y y 1 = m(x x 1 ) y = (x ( 6)) y = (x + 6) y = x + + y = x Plug values into point slope formula Distribute slope Solve for y An important thing to observe about the point slope formula is that the operation between the x s and y s is subtraction. This means when you simplify the signs you will have the opposite of the numbers in the point. We need to be very careful with signs as we use the point-slope formula. In order to find the equation of a line we will always need to know the slope. If we don t know the slope to begin with we will have to do some work to find it first before we can get an equation. Example. Find the equation of the line through the points (,) and (, ). m = y y 1 x x 1 m = ( ) = 8 6 = y y 1 =m(x x 1 ) y = (x ( )) y = (x +) First we must find the slope Plug values in slope formula and evaluate With slope and either point, use point slope formula Example. Find the equation of the line through the points (, ) and ( 1, ) in slope-

3 intercept form. m = y y 1 x x 1 First we must find the slope m= 1 ( ) = 6 = Plug values in slope formula and evaluate y y 1 = m(x x 1 ) With slope and either point, point slope formula y = (x ( )) y = (x + ) y = x 9 Distribute slope Solve for y + + Add to both sides y = x Example 6. Find the equation of the line through the points (6, ) and (, 1) in slopeintercept form. m = y y 1 x x 1 m= 1 ( ) 6 = 10 = 10 y y 1 =m(x x 1 ) First we must find the slope y ( ) = (x 6) 10 y += (x 6) Distribute slope 10 y += 10 x y = 10 x 1 Plug values into slope formula and evaluate Use slope and either point, use point slope formula Solve for y. Subtractfrom both sides Using 10 on right so we have a common denominator World View Note: The city of Konigsberg (now Kaliningrad, Russia) had a river that flowed through the city breaking it into several parts. There were 7 bridges that connected the parts of the city. In 17 Leonhard Euler considered the question of whether it was possible to cross each bridge exactly once and only once. It turned out that this problem was impossible, but the work laid the foundation of what would become graph theory. Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution.0 Unported License. (

4 . Practice - Point-Slope Form Write the point-slope form of the equation of the line through the given point with the given slope. 1) through(, ), slope = undefined ) through(1, ), slope = undefined ) through(, ), slope = 1 ) through( 1, ), slope = 9 7) through(,1), slope= 9) through(0, ), slope = ) through(, 1), slope = 1 6) through(, ), slope = 8) through(, ), slope = 10) through( 1,1), slope= 11) through(0, ), slope = 1 1) through(0, ), slope = 1) through(, ), slope= 1 1) through( 1, ), slope= 1) through( 1,), slope= 16) through(1, ), slope = Write the slope-intercept form of the equation of the line through the given point with the given slope. 17) through: ( 1, ), slope = 18) through: (, ), slope = 19) through: (, 1), slope = 0) through: (, ), slope = 1) through: (, 1), slope = 1 ) through: (, ), slope = 7 ) through: (, ), slope = ) through: (, 0), slope = ) through: (, ), slope = 7) through: (, ), slope =1 9) through:(, ), slope=undefined 6) through: (,), slope= 7 8) through: (, ), slope = 0 0) through: (, ), slope = 1) through: (, ), slope = 1 ) through: (,), slope= 6

5 Write the point-slope form of the equation of the line through the given points. ) through: (, ) and (, 1) ) through: (,1) and (, 0) 7) through: (, ) and (0, ) 9) through: (,) and (, ) 1) through: (, ) and (, ) ) through: (1,) and (, ) 6) through: (, ) and (, ) 8) through: (, 1) and (, ) 0) through: ( 1, ) and (,0) ) through: ( 1, ) and (, ) Write the slope-intercept form of the equation of the line through the given points. ) through: (, 1) and ( 1, ) ) through: (, ) and (, ) 7) through: (,1) and (1,) 9) through: (0,) and (, ) 1) through: (0,) and ( 1, 1) ) through: (, 1) and (, ) 6) through: (1, 1) and (, ) 8) through: (0,1) and (, 0) 0) through: (0,) and (,) ) through: (, 0) and (, ) Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution.0 Unported License. (

6 . 1) x = ) x =1 ) y = 1 (x ) ) y 1 = 1 (x ) ) y + =9(x+1) 6) y + = (x ) 7) y 1 = (x +) 8) y + = (x ) 9) y + = x 10) y 1 =(x + 1) 11) y + = 1 x 1) y = x 1) y + = 1 (x +) 1) y + = (x +1) 1) y = (x +1) 16) y + = x(x 1) 17) y = x 18) y = x + Answers - Point-Slope Form 19) y = x + 0) y = x 10 1) y = 1 x + ) y = 7 x + ) y = x + ) y = x ) y = x 6) y = 7 x 7) y =x 8) y = 9) x = 0) y =x 1 1) y = 1 x ) y = 6 x ) y = (x +) ) y = ) y 1= 1 (x ) 8 6) y = 1 (x + ) 8 7) y += (x + ) 8) y 1= (x +) 8 9) y = 1 (x ) 0) y += (x +1) 1) y += 8 (x ) 7 ) y += 1 (x + 1) ) y = x 11 ) y = 1 10 x ) y = 8 7 x 7 6) y = 1 x 7) y = x+ 8) y = 1 x + 1 9) y = x+ 0) y =x+ 1) y =x + ) y = 7 x Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution.0 Unported License. ( 6