Section 1.5: Point-Slope Form

Transcription

1 Section 1.: 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 (easily) use the slope intercept equation, so we will use a dierent more exible formula. If we let the slope of a line be m, and a specic point on the line be (x 1 ; y 1 ), and any other point on the line be (x; y), then 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 x x 1 ) 1 If we know the slope, m of a line 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 point-slope formula. Point Slope Formula: Example. Write the equation of the line through the point (; ) with a slope of. y ( ) = (x ) y + = (x ) Plug values into point slope formula Often, we will prefer nal answers be written in slope-intercept form. If the directions ask for the answer in slope-intercept form we will simply distribute the slope, then solve for y. 8

2 Example. Write the equation of the line through the point ( 6; ) with a slope of slope-intercept form. in 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 nd 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 nd it rst 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 = (x ( )) y = (x + ) First we must nd 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 slopeintercept form. m = y y 1 x x 1 First we must nd the slope m = 1 ( ) = 6 = Plug values in slope formula and evaluate 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 9

3 Example 6. Find the equation of the line through the points (6; ) and ( ; 1) in slope-intercept form. m = y y 1 x x 1 m = 1 ( ) 6 = 10 = 10 y ( ) = (x 6) 10 y + = (x 6) 10 y + = 10 x y = 10 x 1 First we must nd the slope Plug values into slope formula and evaluate Use slope and either point; use point slope formula Distribute slope Solve for y: Subtract from both sides Using 10 on right so we have a common denominator 0

4 1. Practice Write the point-slope form of the equation of the line through the given point with the given slope. 1) through (; ); slope =undened ) through (1; ); slope =undened ) 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=undened 6) through: (; ); slope = 7 8) through: ( ; ); slope =0 0) through: ( ; ); slope = 1) through: ( ; ); slope = 1 ) through: (; ); slope = 6 1

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 (; )

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 + 1. Answers 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 + 6 7