Unit 2B

Derivatives Part 2

Lesson 3: Finding Tangent Lines Along Curves

Skill Builder

Slope-Point Form of a Linear Equation

The third form of a linear equation you will look at in detail is the slope-point form. This form comes directly from rearranging the slope formula. Remember that in the slope formula, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«mo»,«/mo»«mo»§#160;«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfenced»«/mstyle»«/math» is a point and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»,«/mo»«mo»§#160;«/mo»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«/mrow»«/mfenced»«/mstyle»«/math» is another point.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»m«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mrow»«mrow»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd»«mi»m«/mi»«mfenced»«mrow»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mrow»«menclose notation=¨updiagonalstrike¨»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msup»«mi»x«/mi»«mn»1«/mn»«/msup»«/menclose»«/mfrac»«menclose notation=¨updiagonalstrike¨»«mfenced»«mrow»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfenced»«/menclose»«/mtd»«/mtr»«mtr»«mtd»«mi»m«/mi»«mfenced»«mrow»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»

Often this form is written using «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math» instead of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msub»«mi»x«/mi»«mn»2«/mn»«/msub»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» instead of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msub»«mi»y«/mi»«mn»2«/mn»«/msub»«/mstyle»«/math» , making the formula «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»m«/mi»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfenced»«mo»=«/mo»«mi»y«/mi»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mrow»«/mstyle»«/math».

Slope-Point Form
a linear equation of the form «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»m«/mi»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfenced»«mo»=«/mo»«mi»y«/mi»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mstyle»«/math» , where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»m«/mi»«/mstyle»«/math» represents the slope of the graph of the corresponding linear relation

As the name implies, the slope-point form of a linear equation is most useful when you know the slope, along with a point on the line. Although any point from the graph of the relation can be used with this form, it is often helpful to pick points that are easy to work with, such as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»1«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math» or «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»5«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»0«/mn»«/mrow»«/mfenced»«/mstyle»«/math», if possible.

Determine the slope-point equation of a line with a slope of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«/mrow»«/mstyle»«/math» that passes through the point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»4«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mstyle»«/math».

Enter the point and the slope into the slope-point equation.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»m«/mi»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«mo»-«/mo»«mfenced»«mrow»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»4«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«mo»+«/mo»«mn»1«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mfrac»«mn»2«/mn»«mn»3«/mn»«/mfrac»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mn»4«/mn»«/mrow»«/mfenced»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
The equation «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»-«/mo»«mn»4«/mn»«mo»=«/mo»«mn»2«/mn»«mo».«/mo»«mn»7«/mn»«mfenced»«mrow»«mi»x«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math» represents a linear relation. State the slope of the graph of this relation and a point you know will be on the graph of the relation.

The equation is in slope-point form, so the slope can be determined by inspection.

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»m«/mi»«mo»=«/mo»«mn»2«/mn»«mo».«/mo»«mn»7«/mn»«/mrow»«/mstyle»«/math»

To interpret this equation correctly, it may help to write the addition as a subtraction of a negative. Doing so will help the equation better resemble the form «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»-«/mo»«msub»«mi»y«/mi»«mn»1«/mn»«/msub»«mo»=«/mo»«mi»m«/mi»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«msub»«mi»x«/mi»«mn»1«/mn»«/msub»«/mrow»«/mfenced»«/mrow»«/mstyle»«/math».

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«mo»-«/mo»«mn»4«/mn»«mo»=«/mo»«mi»m«/mi»«mfenced»«mrow»«mi»x«/mi»«mo»-«/mo»«mfenced»«mrow»«mo»-«/mo»«mn»3«/mn»«/mrow»«/mfenced»«/mrow»«/mfenced»«/mstyle»«/math»

The point «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mo»-«/mo»«mn»3«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»4«/mn»«/mrow»«/mfenced»«/mstyle»«/math» lies on the graph of the relation.
In the equation of the line defined by «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»m«/mi»«mi»x«/mi»«mo»+«/mo»«mi»b«/mi»«/mrow»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»m«/mi»«/mstyle»«/math» represents the slope of the line and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»b«/mi»«/mstyle»«/math» represents the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-intercept.

Slope-Intercept Form
a linear equation of the form «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mi»m«/mi»«mi»x«/mi»«mo»+«/mo»«mi»b«/mi»«/mrow»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»m«/mi»«/mstyle»«/math» represents the slope and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»b«/mi»«/mstyle»«/math» represents the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-intercept of the graph of the corresponding linear relation

«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-intercept
the point at which the graph of a relation crosses the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-axis

The «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-intercept can be represented by the point of intersection between the graph and the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-axis or by just the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-value of that point of intersection. Both of the following statements represent the same information.

  • The «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-intercept occurs at «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mfenced»«mrow»«mn»0«/mn»«mo»,«/mo»«mo»§#160;«/mo»«mn»4«/mn»«/mrow»«/mfenced»«/mstyle»«/math».
  • The «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-intercept is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»4«/mn»«/mstyle»«/math».

The «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-intercept is more generally known as the vertical-axis intercept. Although «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math» is a common label for the vertical axis, other variables can be used. The graph below shows a «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»d«/mi»«/mstyle»«/math»-intercept.







General Form of a Linear Equation

It was advantageous to work with the slope-intercept form of a linear equation because you were able to readily identify the slope and the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»y«/mi»«/mstyle»«/math»-intercept of the graph of the related linear relation. There are other ways of writing linear equations that offer different advantages. One of these other ways is called the general form.

General Form
the equation of a line written as «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»A«/mi»«mi»x«/mi»«mo»+«/mo»«mi»B«/mi»«mi»y«/mi»«mo»+«/mo»«mi»C«/mi»«mo»=«/mo»«mn»0«/mn»«/mrow»«/mstyle»«/math», where «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»A«/mi»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»B«/mi»«/mstyle»«/math» are not both zero; by convention, «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»A«/mi»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»B«/mi»«/mstyle»«/math», and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»C«/mi»«/mstyle»«/math» are whole numbers, and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»A«/mi»«/mstyle»«/math» is positive

Express «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/mrow»«/mstyle»«/math» in general form.


«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»y«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mi»x«/mi»«mo»+«/mo»«mn»4«/mn»«/mtd»«/mtr»«mtr»«mtd»«mi»y«/mi»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mi»x«/mi»«mo»-«/mo»«mn»4«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«menclose notation=¨updiagonalstrike¨»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mi»x«/mi»«/menclose»«menclose notation=¨updiagonalstrike¨»«mo»+«/mo»«mn»4«/mn»«/menclose»«mo»+«/mo»«menclose notation=¨updiagonalstrike¨»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mi»x«/mi»«/menclose»«menclose notation=¨updiagonalstrike¨»«mo»-«/mo»«mn»4«/mn»«/menclose»«/mtd»«/mtr»«mtr»«mtd»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«mo»-«/mo»«mn»4«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«mtr»«mtd»«mn»6«/mn»«mfenced»«mrow»«mfrac»«mn»1«/mn»«mn»6«/mn»«/mfrac»«mi»x«/mi»«mo»+«/mo»«mi»y«/mi»«mo»-«/mo»«mn»4«/mn»«/mrow»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»6«/mn»«mfenced»«mn»0«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd»«mi»x«/mi»«mo»+«/mo»«mn»6«/mn»«mi»y«/mi»«mo»-«/mo»«mn»24«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»0«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
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