Point-Slope Form

Point-Slope Form exercise

• Describe the point-slope form and the slope formula.

  Hints

  There's also the slope-intercept form:

  $y=mx+b$, where

  • $m$ is the slope and
  • $b$ is the y-intercept.

  The slope is the change in $y$ divided by the change in $x$.

  Each point in the coordinate system is defined by the x-coordinate and the y-coordinate.

  Solution

  Here we have two different equations:

  The point-slope form

  $y-y_1=m(x-x_1)$

  For this equation, we have to know one point $(x_1,y_1)$ on the line and the slope $m$.

  The slope formula

  $m=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{x_2-x_1}$

  In order to use this formula, we have to know two points $(x_1,y_1)$ and $(x_2,y_2)$. Now we can calculate the slope by dividing the change in $y$ by the change in $x$.

• Determine the slope of the equation.

  Hints

  Use the correct order of subtraction.

  The slope shows the increasing number of logs for each month starting with 20 logs in January.

  Solution

  From the given information, we know get two points:

  • $(0,20)$ and
  • $(10,180)$.

  This enables us to use the slope formula

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

  Now we plug in the coordinates of the points. It's very important to be consistent with the order of subtraction:

  $m=\frac{180-20}{10-0}=\frac{160}{10}=16$.

• Determine the slope and the coordinates needed for the point-slope form.

  Hints

  The variable $x$ corresponds to the number of months. $y$ corresponds to the number of logs.

  The slope is the increase of the number of logs over the number of months.

  Solution

  At the beginning, $x_1=0$, there are 30 logs, $y_1=30$.

  The beavers work 11 months, giving us a second x coordinate, $x_2=11$. The number of logs at this point, $y_2$, is unknown.

  Each month they chop down 15 trees and produce 15 logs a month. This gives us the slope $m$.

  Now we can write the equation:

  $y-y_1=m(x-x_1)$

  Let's plug in the values:

  $y-30=15(x-0)=15x$.

  After adding $30$ we can write the equation as $y=15x+30$.

  If we plug $x_2=11$ into this equation, we get the amount of logs in November:

  $y=15\times 11+30=195$.

  Busy beavers!

• Decide which equation corresponds to which line.

  Hints

  The points in the graph are

  • $P(3,60)$,
  • $Q(8,60)$
  • $R(8,50)$

  You can determine the slope by counting

  • the number of steps to the right and
  • the corresponding number of steps up or down.

  All the slopes are positive.

  Remember that the x-axis and the y-axis have different scales.

  Solution

  The point-slope form is:

  $y-y_1=m(x-x_1)$.

  So we need:

  • one point $(x_1,y_1)$
  • the slope

  The red line

  • $P(3,60)$
  • slope: one step to the right and twenty steps up $m=\frac{20}1=20$.

  This gives us:

  $y-60=20(x-3)$.

  The green line

  • $Q(8,60)$
  • slope: two steps to the right and ten steps up $m=\frac{10}2=5$.

  This gives us:

  $y-60=5(x-8)$.

  The blue line

  • $Q(8,50)$
  • slope: four steps to the right and ten steps up $m=\frac{10}4=\frac52$.

  This gives us:

  $y-50=\frac52(x-8)$.

• Decide which equation is in point-slope form.

  Hints

  In order to solve the point-slope form we need:

  • one point
  • the slope.

  Any point in the coordinate system has

  • an x-coordinate and
  • a y-coordinate.

  The slope has to be multiplied by $x$.

  There's also the slope-intercept form:

  $y=mx+b$.

  Solution

  The point-slope form can be solved with

  • a point $(x_1,y_1)$ and
  • the slope $m$.

  $y-y_1=m(x-x_1)$.

• Determine how many logs each beaver business has at the end of the year.

  Hints

  Write down the point-slope form for each business.

  The point-slope form is:

  $y-y_1=m(x-x_1)$.

  The first coordinate of a point is the x-coordinate, the second is the y-coordinate.

  One year later is represented by $x_2=12$.

  Solution

  We can plug the values into the point-slope form

  $y-y_1=m(x-x_1)$.

  Justin's beaver business: $m=20$, $(0,35)$.

  $y-35=20(x-0)=20x$.

  After adding $35$ rewrite the equation as $y=20x+35$.

  Now we plug in $x_2=12$ for one year:

  $y=20\times 12+35=275$.

  Paul*s beaver business: $m=15$, $(0,42)$.

  $y-42=15(x-0)=15x$.

  After adding $42$ rewrite the equation as $y=15x+42$.

  Now we plug in $x_2=12$ for one year:

  $y=15\times 12+42=222$.

  Adele's beaver business: $m=25$, $(0,5)$.

  $y-5=25(x-0)=25x$.

  After adding $5$ rewrite the equation as $y=25x+5$.

  Now we plug in $x_2=12$ for one year:

  $y=25\times 12+5=305$.

  Janet's beaver business: $m=18$, $(0,12)$.

  $y-12=18(x-0)=18x$.

  After adding $12$ rewrite the equation as $y=18x+12$.

  Now we plug in $x_2=12$ for one year:

  $y=18\times 12+12=228$.