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3. Linear Functions
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Algebra 1

3.05 Point-slope form

Lesson
Practice

Point-slope form

When we are given the coordinates of a point on the line and the slope of that line, then the point-slope form can be used to state the equation of the line.

Coordinates can be given as an ordered pair in a table of values, read from a graph, or described in a scenario. The slope can be stated as a value, calculated from two points, read from a graph, or given as a rate of change in a scenario.

Point-slope form of a linear relationship is represented by:

\displaystyle y-y_{1}=m\left(x-x_{1}\right)
\bm{m}
slope
\bm{x_{1}}
x-coordinateof a point on the line
\bm{y_{1}}
y-coordinate of the same point
-4
-3
-2
-1
1
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4
x
-4
-3
-2
-1
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y

This graph shows the point-slope function

y - 4 = \dfrac{1}{2}(x - 2)

We see that the line has a slope of \dfrac{1}{2} and contains the point (2,\,4).

-4
-3
-2
-1
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x
-4
-3
-2
-1
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y

The same function could also be represented as

y - 2 = \dfrac{1}{2}(x + 2)

since the slope remains constant at \dfrac{1}{2} but uses the point on the line (-2,\,2).

We can also find the equation in point-slope form when given the coordinates of two points on the line by following these steps:

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-9
-8
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-6
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y

1. Choose which ordered pair will represent (x_{1},\,y_{1}) and the other (x_{2},\,y_{2}).

x_{1} = 3,\,y_{1}=5

x_{2} = 7,\,y_{2}=-2

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y

2. Solve for slope using m =\dfrac{y_{2} - y_{1}}{x_{2} - x_{1}}.

\begin{aligned}m&=\dfrac{-2-5}{7-3}\\&=-\dfrac{7}{4}\end{aligned}

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y

3. Rewrite y - y_{1} = m(x - x_{1}), substituting m and (x_{1},\, y_{1}).

y-5 = -\dfrac{7}{4}(x-3)

We are also able to find the slope-intercept form of an equation written in point-slope form by solving the equation for y.

Examples

Example 1

Given the point (4,\,-1) and the slope m = 2, write the equation of the line in point-slope form.

Worked Solution
Create a strategy

To write the equation of the line in point-slope form, we use the formula: y - y_1 = m(x - x_1) where (x_1,\,y_1) is a point on the line and m is the slope.

Apply the idea
\displaystyle y - (-1)\displaystyle =\displaystyle 2(x - 4)Substitute x_1 = 4,\,y_1 = -1,\, and m = 2
\displaystyle y + 1\displaystyle =\displaystyle 2(x - 4)Simplify

The equation of the line in point-slope form is y + 1 = 2(x - 4).

Example 2

The graph of a linear function is shown. Write the equation of the line in point-slope form for each of the given points on the line.

-6
-5
-4
-3
-2
-1
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x
-6
-5
-4
-3
-2
-1
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y
a

(-6,\,1)

Worked Solution
Apply the idea

From the graph, we can choose any two points to find the slope. We can either count the \dfrac{\text{rise}}{\text{run}} from the graph or use the slope formula \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}}.

\displaystyle m\displaystyle =\displaystyle \dfrac{y_{2} - y_{1}}{x_{2} - x_{1}}Start with the slope formula
\displaystyle =\displaystyle \dfrac{-3 - 1}{6 - (-6)}Substitute (-6,\,1) = (x_{1},\ ,y_{1}) and (6,\,-3) = (x_{2},\, y_{2})
\displaystyle =\displaystyle \dfrac{-4}{12}Evaluate the subtraction
\displaystyle =\displaystyle -\dfrac{1}{3}Simplify

Find the equation of the line in point-slope form:

\displaystyle y - y_{1}\displaystyle =\displaystyle m(x - x_{1})Start with the point-slope form
\displaystyle y-1\displaystyle =\displaystyle -\dfrac{1}{3}(x+6)Substitute y_{1}=1,\,m=-\dfrac{1}{3},\, and x_{1}=-6
b

(3,\,-2)

Worked Solution
Apply the idea

The slope from the previous part was m = -\dfrac{1}{3} and will use (3,\,-2) as (x_{1},\, y_{1}).

Given y - y_{1} = m(x - x_{1}), the resulting equation will be y + 2 = -\dfrac{1}{3}(x - 3)

Example 3

A line passes through the two points \left(-3,\,7\right) and \left(2,\,-3\right).

a

Write the equation of the line in point-slope form.

Worked Solution
Create a strategy

We will first find the slope of the line using the two points. Then we will pick one of the points to substitute into the point-slope equation.

Apply the idea

Find the slope of the line:

\displaystyle m\displaystyle =\displaystyle \dfrac{y_{2}-y_{1}}{x_{2}-x_{1}}Start with the slope formula
\displaystyle =\displaystyle \dfrac{-3-7}{2-\left(-3\right)}Substitute \left(x_{1},\,y_{1}\right)=\left(-3,\,7\right) and \left(x_{2},\,y_{2}\right)=\left(2,\,-3\right)
\displaystyle =\displaystyle \dfrac{-10}{5}Evaluate the subtraction
\displaystyle =\displaystyle -2Simplify

Find the equation of the line in point-slope form:

\displaystyle y-y_1\displaystyle =\displaystyle m\left(x-x_{1}\right)Start with the point-slope formula
\displaystyle y-7\displaystyle =\displaystyle -2\left(x-\left(-3\right)\right)Substitute y_{1}=7,\, m=-2,\, and x_{1}=-3
\displaystyle y-7\displaystyle =\displaystyle -2\left(x+3\right)Simplify
Reflect and check

We can confirm we have correctly written this in point-slope form: y-y_{1}=m\left(x-x_{1}\right), as we can see the coordinates \left(-3,\,7\right), and a slope of -2 are represented correctly. This is easier to see in the previous line: y-7=-2\left(x-\left(-3\right)\right)

b

Determine whether the ordered pair (-10,\,21) lies on the same line as \left(-3,\,7\right) and \left(2,\,-3\right).

Worked Solution
Create a strategy

We can substitute the ordered pair into the equation we wrote in part (a) and determine whether the statement is true. If so, we can confirm whether the point lies on the same line as \left(-3,\,7\right) and \left(2,\,-3\right).

Apply the idea

We have

\displaystyle y-7\displaystyle =\displaystyle -2(x+3)Point-slope form from part (a)
\displaystyle (21)-7\displaystyle =\displaystyle -2((-10)+3)Substitute y=21 and x=-10
\displaystyle 14\displaystyle =\displaystyle -2(-7)Evaluate the subtraction and addition
\displaystyle 14\displaystyle =\displaystyle 14Evaluate the multiplication

Since the resulting equation is true, the ordered pair (-10,\, 21) satisfies the equation and as a result is on the same line as \left(-3,\,7\right) and \left(2,\,-3\right).

Reflect and check

Any ordered pair that satisfies the equation will be on the same line as \left(-3,\,7\right) and \left(2,\,-3\right).

Example 4

A carpenter charges for a day's work using the given equation, where y is the cost and x is the number of hours worked:y-125=50\left(x-2\right)

a

Interpret the meaning of each number in the equation.

Worked Solution
Create a strategy

Recall the point-slope form: y-y_{1}=m\left(x-x_{1}\right)

Apply the idea

The slope is represented by the number 50. In this context this is the rate the carpenter charges for one hour of work. So, the carpenter charges \$50 for each hour of work.

Using the point slope formula, we also know the point \left(2, 125 \right) fits our context. This means after 2 hours of work, the total cost is \$125.

b

Draw the graph of the linear equation from the point-slope form. Clearly label the axes with labels, units, and an appropriate scale.

Worked Solution
Create a strategy

We need two points to plot a line. We can read from the equation that one point on the line is \left(2,\,125\right). To get another point, we can use the slope from the given point or substitute in an x-value in the domain and solve for y.

Since the x-axis will be the number of hours worked and generally an 8-hour work day is reasonable, showing the graph for 0\leq x\leq 8 would be a good scale.

Apply the idea

Before we graph, we label our axes. To do that, we need to know what our maximum and minimum values should be.

We can use the slope or the equation to determine the y-value when x=8. We can find that the y-value when x=8 is y=425.

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\text{Hours }(x)
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450
\text{Charge in }\$ \,(y)

A reasonable scale for the x-axis from 0 to 8 going up by 1.

A reasonable scale for the y-axis from 0 to 450 going up by 50 with one tick between each label at the 25s.

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\text{Hours } (x)
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\text{Charge in }\$ \,(y)

The given point is \left(2,\,125\right).

Since the slope is 50, we can go up 50 units and right 1 unit from our given point to plot another point on the line.

c

Predict the charge for 6 hours of work using the graph.

Worked Solution
Create a strategy

We can go to x=6 on the x-axis and then go up to the line and across to the y-axis to make our prediction.

Apply the idea
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\text{Hours } (x)
50
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\text{Charge in }\$ \,(y)

The charge for 6 hours of work will be \$ 325.

d

Give an example of a non-viable solution if the carpenter only uses this model for a maximum of 10 hours per day. Explain your answer.

Worked Solution
Create a strategy

Since the carpenter only uses this model between 0 and 10 hours, any solution outside of this interval will not be viable.

Apply the idea

One possible non-viable solution would be \left(11,\, 575\right) which would represent working 11 hours and getting paid \$575, but the model only applies to a maximum of 10 hours, so we don't actually know what would happen for 11 hours.

Reflect and check

Any solution with x \lt 0 or x \gt 10 would be non-viable.

Idea summary

The point-slope form of a line is:

\displaystyle y-y_{1}=m\left(x-x_{1}\right)
\bm{m}
slope
\bm{x_{1}}
x-coordinate of a point on the line
\bm{y_{1}}
y-coordinate of the same point

Point-slope form is useful when we know or want to know the slope of the line and a point on the line.

Outcomes

A.F.1

The student will investigate, analyze, and compare linear functions algebraically and graphically, and model linear relationships.

A.F.1a

Determine and identify the domain, range, zeros, slope, and intercepts of a linear function, presented algebraically or graphically, including the interpretation of these characteristics in contextual situations.

A.F.1c

Write equivalent algebraic forms of linear functions, including slope-intercept form, standard form, and point-slope form, and analyze and interpret the information revealed by each form.

A.F.1di

Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: i) given the graph of a line;

A.F.1dii

Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: ii) given two points on the line whose coordinates are integers;

A.F.1diii

Write the equation of a linear function to model a linear relationship between two quantities, including those that can represent contextual situations. Writing the equation of a linear function will include the following situations: iii) given the slope and a point on the line whose coordinates are integers;

A.F.1f

Graph a linear function in two variables, with and without the use of technology, including those that can represent contextual situations.

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