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4.06 Point-slope form

Lesson

Guiding questions

As you explore the applet and read the lesson below, think about your answers to the following questions.  Discuss your answers with a partner after reading the lesson.

  1. If we are given a point on the line and the slope of the line, what is the best way to find the equation of the line? 
  2. In what ways are the slope formula and the point-slope form connected to one another?

 

 

So far we have seen two different forms of the equation for a straight line.  

Equations of lines

We have:

$y=mx+b$y=mx+b  (slope intercept form)

$Ax+By=C$Ax+By=C   , where $A$A, $B$B, $C$C are integers and $A>0$A>0 (standard form)

If we are given a point on the line and the slope of the line, what is the best way to find the equation of the line?  

We have a couple of options. 

Method 1 - Using $y=mx+b$y=mx+b

We could use this information and construct an equation in slope-intercept form.

Worked example

Find the equation of the line that passes through the point $\left(2,-8\right)$(2,8) and has slope of $-2$2

Think:  We can instantly identify the $m$m value in $y=mx+b$y=mx+b$m=-2$m=2

If the point $\left(2,-8\right)$(2,8) is on the line, then it will satisfy the equation, so we can substitute it in to find the value of $b$b.

Do$y=mx+b$y=mx+b

$y=-2x+b$y=2x+b

To find $b$b: we can substitute the values of the point $\left(2,-8\right)$(2,8)

$-8=-2\times2+b$8=2×2+b  

and we can now solve for $b$b

$-8=-4+b$8=4+b

$-4=b$4=b

Reflect: So the equation of the line is $y=-2x-4$y=2x4

 

Method 2 - The point-slope formula

We are going to derive a new equation for a line using what we know about the equation for slope.

Worked example

Let's do the same question as we did for Method 1, so we want to find the equation of the line that passes through the point $\left(2,-8\right)$(2,8) and has slope of $-2$2

Think: Using the same values as the question in Method 1, we know that the slope of the line is $-2$2. We also know a point on the line, $\left(2,-8\right)$(2,8). Now, apart from this point there are infinitely many other points on this line, and we will let $\left(x,y\right)$(x,y) represent each of them.

Well, since $\left(x,y\right)$(x,y) and $\left(2,-8\right)$(2,8) are points on the line, then the slope between them will be $-2$2.

We know that to find the slope given two points, we use: 

$m=\frac{y_2-y_1}{x_2-x_1}$m=y2y1x2x1

Do: Let's apply the slope formula to $\left(x,y\right)$(x,y) and $\left(2,-8\right)$(2,8):

$m=\frac{y-\left(-8\right)}{x-2}$m=y(8)x2

But we know that the slope of the line is $-2$2. So:

$\frac{y-\left(-8\right)}{x-2}=-2$y(8)x2=2

Rearranging this slightly, we get:

$y-\left(-8\right)=-2\left(x-2\right)$y(8)=2(x2)

You may be thinking that we should simplify the equation, and of course if you do you should get $y=-2x-4$y=2x4. What we want to do though, is generalize our steps so that we can apply it to any case where we're given a slope $m$m, and a point on the line $\left(x_1,y_1\right)$(x1,y1).

In the example above, the point on the line was $\left(2,-8\right)$(2,8). Let's generalize and replace it with $\left(x_1,y_1\right)$(x1,y1). We were also given the slope $-2$2. Let's generalize and replace it with $m$m.

$y-\left(-8\right)=-2\left(x-2\right)$y(8)=2(x2)      becomes     $y-y_1=m\left(x-x_1\right)$yy1=m(xx1)

 

The point-slope formula

Given a point on the line $\left(x_1,y_1\right)$(x1,y1) and the slope $m$m, the equation of the line is:

$y-y_1=m\left(x-x_1\right)$yy1=m(xx1)

 

Worked example

Find the equation of a line that passes through the point $\left(-4,3\right)$(4,3) and has slope of $5$5

$y-y_1$yy1 $=$= $m\left(x-x_1\right)$m(xx1) Use the point-slope formula
$y-3$y3 $=$= $5\left(x-\left(-4\right)\right)$5(x(4)) Fill in the given values
$y-3$y3 $=$= $5\left(x+4\right)$5(x+4) Simplify
$y-3$y3 $=$= $5x+20$5x+20  
$y$y $=$= $5x+23$5x+23 Re-arrange to slope-intercept form

A much tidier method than the method used in the previous example!

 

Practice questions

question 1

A line passes through the point $A$A$\left(-2,-9\right)$(2,9) and has a slope of $-2$2. Using the point-slope formula, express the equation of the line in slope intercept form.

question 2

A line passes through the point $A$A$\left(\frac{7}{8},-5\right)$(78,5) and has a slope of $2$2. Using the point-slope formula, express the equation of the line in slope intercept form.

Question 3

A line has slope $2$2 and passes through the point ($-1$1, $5$5).

  1. By substituting into the equation $y=mx+b$y=mx+b, find the value of $b$b for this line.

  2. Now write the equation of the line in slope-intercept form.

  3. Derive the same equation by substituting into the point-slope formula.

  4. Graph the line.

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Outcomes

I.F.IF.4

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior.

I.F.IF.7

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

I.F.IF.7.a

Graph linear functions and show intercepts.

I.F.LE.2

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

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