SlopeIntercept Form and PointSlope Form


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1 SlopeIntercept Form and PointSlope Form In this section we will be discussing SlopeIntercept Form and the PointSlope Form of a line. We will also discuss how to graph using the SlopeIntercept Form. Slopeintercept form: When an equation is in slopeintercept form we can directl read the slope and the intercept from the equation. We can also graph immediatel using this information. Slopeintercept form: = m + b where m is the slope and (0,b) is the intercept. Eample: = + We have slope of and intercept (0,) Note that the purple square is the intercept (0,) and red points are found b using the slope of Remember that we are seeing a visual of all the solutions to the equation (ever point on the line is a solution).
2 PointSlope form: When an equation is in pointslope form, we can directl read the slope and a point that the line passes through from the equation. Usuall we don t graph from this form because we don t often see problems written this wa, but we can. = m or = m + where m is the slope and Pointslope form: ( ) ( ) (, ) is a point on the line. Eample: = ( ) or = ( ) + Here the slope is and the point that the line passes through is (,). Note that when given in the first form = ( ) the subtraction signs do not affect the coordinates of the points Note that the purple square is the point given b the equation (,) and red points are found b using the slope of. You might have noticed that the graphs are the same. If we solve the equation given in pointslope form for, we end up with the equation in slopeintercept form: ( ) = = = + Standard form: Standard form is not a ver helpful form as far as graphing, but some books ask students to write their equations in standard form. Your instructor ma want ou to write our work in slopeintercept form instead.
3 Standard Form: A + B = C where A,B and C are real numbers and A and B can not both equal 0 at the same time. We have seen the graph of = + above. In standard form, this equation would look like  + =. However, this is not the onl wa we could write it. Often books will have A positive. We can do this if we multipl both sides of the equation b : = . There are multipl was to write an equation in Standard Form and it still be correct. So, if our book writes the answers in Standard Form, ou ma need to manipulate their equation to see if it matches what ou found. Since there are multipl was to write Standard Form, man instructors prefer students write their lines in slopeintercept form. Notes about slopeintercept form:. The slope is a number, so don t write a variable when ou are identifing m.. Write the intercept as a point. Since it is a point on the ais, the coordinate will be 0.. You MUST solve for to find the slope and intercept. = + is not slopeintercept form, and the slope is NOT and the intercept is NOT (0,).. + = is not in slopeintercept form. We must rewrite with the under each of the terms in the numerator: intercept is (0,). = +, which simplifies to = + ; So, m = and the  5. When graphing using the slope intercept form, first plot the intercept. Then use the slope to find additional points. Note that a slope of means we would rise up in the positive direction units and run to the right in the positive direction units. We can repeat this as often as we want. Also, since is the same as, we could also rise down in the negative direction units and run to the left in the negative direction units and be on the same line. Similarl for slopes such as 5, we can associate the negative with the
4 numerator and denominator to get additional points going both directions on the line: = =. EXAMPLE : Put  + = into slopeintercept form, find the slope and the intercept (write as a point). Graph the equation using the  intercept and the slope. Plot the intercept point and three additional points using the slope. Solution: First we need to put + = slopeintercept form b solving for : + = = + = + = + The slope is m = and the intercept is (0,) Note the we have labels on the aes, the scale on both the aes, and the arrows on the line. Also note how the slope gives us points going both was on the line from the intercept and that the line rises from left to right as it should with a positive slope.
5 Finding equations of lines given various information: To find the equation of a nonvertical line, we need onl pieces of information:. The slope.. A point on the line. You ma use either the slopeintercept form or the pointslope form to find equations of lines. The eample below shows both forms. Pick our favorite and practice using it. Note: If finding the equation for a horizontal or vertical line, it ma be easiest to graph it to find the equation! EXAMPLE : Find the equation of a line in slopeintercept form (if possible) for the line that passes through the points (, ) and (6, 5) and write it as a function (if possible) Solution: We need to find the slope since it is not given: 5 m = =. 6 Now we need to find b, the intercept. We can do this two was:. Using the slopeintercept form: = m + b 5 = ( 6 ) + b () 5 = () + b () 5 = + b () = b () So we have = + (5) (): Fillin one of the points into the variables and. It does not matter which point ou pick. Also fillin the slope. ()(): Solve for b. (5): Fillin the slope and the  intercept, but do NOT fillin numbers for and. The and are our variables and MUST sta variables as the represent ever point on the line as ordered pairs (,)! 5
6 . Using the pointslope form: ( ) ( ) = m where, isa point on theline. = ( ) () = () = + ( ) (): Fillin one of the points into the and. It does not matter which point ou pick. Also fillin the slope. ()(): Solve for. Note that we have the same equation for both methods! EXAMPLE : Find the equation of a line in slopeintercept form (if possible) for a line that has undefined slope and passes through the point (, 6). Solution: We alread have a point and a slope. But, the slope isn t one we can fillin to either formula because it isn t a number! Recall that vertical lines have undefined slopes. So, we want a vertical line that passes through the point (,6). Visualize this b making a graph: The equation of this line is =. We can not put it in slopeintercept form because the slope is undefined and there is no intercept. 6
7 EXAMPLE : Find the equation of a line in slopeintercept form (if Solution: possible) that passes through the point (,) and is parallel to the line = +. This problem is often misunderstood b students. Often students think that the line = + passes through the point (, ). It does not. We can verif this b filling in the coordinates of the point into the line:? = ( ) +? = + It ma be helpful to graph what we are given to understand what we want to find =(/) (,)   We want to find the equation of a line that is parallel to the given line and passes through the point (, ). There are infinitel man lines that are parallel to the given line, but onl one that passes through the point (, ). See the graph the pink line is the one we want the equation for. The nosemucus green lines are just eamples of other parallel lines to the original (blue) line. =(/) (,)   7
8 Recall that we onl need two pieces of information to find the equation of a line: () the slope and () a point on the line. We know that the pink line has slope of because it is parallel to the given line = + that has slope of. We also know a point that our pink line will pass through: (,). We can use the slopeintercept form or the pointslope form to find the equation: Slopeintercept form: = m + b = ( ) + b = + b + = b = So the equation of the pink line is b =. Pointslope form: = m ( ) = + = ( + ) + = + ( ) ( ) ( ) = EXAMPLE 5: Let s change the prior eample slightl: Find the equation of a line in slopeintercept form (if possible) that passes through the point (,) and is perpendicular to the line = +. Solution: The onl change to this problem and what we had in the prior eample is now we want to find a line that is perpendicular instead of parallel. Again, it might help to think of this visuall: 8
9 =(/) (,) As before the nosemucus green lines are just random perpendicular lines to the original blue line that we are given. We want to find the equation of the pink line that is perpendicular, but also passes through the point (, ). Recall that we onl need two pieces of information to find the equation of a line: () the slope and () a point on the line. We know that the pink line has slope of  because it is perpendicular to the given line = + that has slope of (recall that perpendicular lines have slopes that are negative reciprocals of each other). We also know a point that our pink line will pass through: (,). We can use the slopeintercept form or the pointslope form to find the equation: Slopeintercept form: = m + b ( ) = + b = + b = b Pointslope form: = m ( ) ( ) ( ) ( ) + = = ( ) = + = + So the equation of the pink line is =. We could have also found this one prett easil b backtracking from the graph. Vertical lines and the slopeintecept form (or the pointslope form): Vertical lines are the onl tpe of line that we can t put into slopeintercept form (or pointslope form). This is because the slope is undefined. We can t write the word undefined for m in = m 9
10 + b! Equations of vertical line will be = the number where the line crosses the ais. Eample: = would be the equation of a vertical line that crosses the ais at. Final comments about lines:. Horizontal lines will have equations of the form = b where b is the intercept (where the line crosses the ais). Note that = b is slopeintercept form with the slope as 0: = 0+b. A line of the form = m is also in slopeintercept form, where the intercept is (0,0). Pros and Cons for using PointSlope Form, m( ) = to find equations of lines: Pro: Man students find it easier and don t make the mistake of fillingin numbers into the variables ( and ) for the final equation that sometimes happens when using the slopeintercept form. Con: You will need to memorize another formula and know when and how to appl it. 0
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