Point-Slope Form of a Line
Emma rents an electric scooter downtown. The app says a 2-hour rental costs $\$9$ and that the price goes up $\$3$ for each extra hour. How much would 5 hours cost? The app never shows the starting fee (the y-intercept), but you don't need it. Point-slope form lets you write a line's equation from any point and the slope.
Building the Point-Slope Form
Suppose a line has slope $m$ and passes through the point $(x_1, y_1)$. Take any other point $(x, y)$ on the line. The slope between the two points is the rise over the run:
$$m = \frac{y - y_1}{x - x_1}$$Multiply both sides by $(x - x_1)$ and you get point-slope form.
Why does it work? Starting at $(x_1, y_1)$, moving $x - x_1$ units to the right makes $y$ change by $m$ times that amount. That is exactly what "slope" means.
Play with the sliders. The point $(h, k)$ is where the line is "pinned," and $m$ tilts it around that point.
You can also read an equation backward to find the slope and a point.
Write the point-slope equation of the line that passes through the point $P(3, -2)$ and has a slope of $m = 4$.
Write the point-slope equation of the line that passes through the point $P(3, -2)$ and has a slope of $m = 4$.
Given the line with equation $y - 5 = -2(x + 1)$, state its slope and obtain the coordinates of a point it passes through directly from the formula.
Given the line with equation $y - 5 = -2(x + 1)$, state its slope and obtain the coordinates of a point it passes through directly from the formula.
Write the point-slope equation of the angle bisector of the first and third quadrants. Then, write the point-slope equation of the angle bisector of the second and…
Write the point-slope equation of the angle bisector of the first and third quadrants. Then, write the point-slope equation of the angle bisector of the second and fourth quadrants, using the origin in both cases.
A Line Through Two Points or from a Graph
Often nobody gives you the slope. You get two points, or a graph. The plan never changes: first find the slope, then use point-slope form with either point.
On a graph, pick two points where the line crosses grid intersections exactly. Count the rise (up is positive, down is negative) and the run (right is positive), then divide.
Calculate the slope of the line that passes through the points $A(-1, 5)$ and $B(2, 11)$. Then, write its equation in point-slope form using the point $A$.
Calculate the slope of the line that passes through the points $A(-1, 5)$ and $B(2, 11)$. Then, write its equation in point-slope form using the point $A$.
Look at the following graph. From the highlighted points, calculate the slope of the line and write its equation in point-slope form using the point $P$ as a reference.
Look at the following graph. From the highlighted points, calculate the slope of the line and write its equation in point-slope form using the point $P$ as a reference.
The following graph shows the constant speed descent profile of a submarine. From the indicated points, determine the point-slope equation of its trajectory.
The following graph shows the constant speed descent profile of a submarine. From the indicated points, determine the point-slope equation of its trajectory.
Converting to Other Forms
One line can be written in several equivalent ways. Point-slope form is easy to write, but sometimes you want a different form.
Some problems call $Ax + By + C = 0$ the general equation of a line. It is the same idea as standard form, just with everything on one side.
Sometimes you are given only this kind of equation and need point-slope form. Choose an $x$-value, find the matching $y$, and you have a point. Pick an $x$ that makes $y$ a whole number so the work stays clean.
Transform the point-slope equation $y + 3 = \frac{1}{2}(x - 4)$ into its slope-intercept form $y = mx + n$.
Transform the point-slope equation $y + 3 = \frac{1}{2}(x - 4)$ into its slope-intercept form $y = mx + n$.
From the general equation of the line $3x - 5y + 15 = 0$, find a point on it whose $x$-coordinate is an integer, calculate its slope and write the corresponding equation…
From the general equation of the line $3x - 5y + 15 = 0$, find a point on it whose $x$-coordinate is an integer, calculate its slope and write the corresponding equation in point-slope form.
Parallel and Perpendicular Lines
Parallel lines never meet, so they must tilt the same way. Perpendicular lines meet at a right angle. Turn a slope triangle $90^\circ$ and the rise and run swap places, and one of them changes sign.
To get a negative reciprocal, flip the fraction and change the sign: the perpendicular slope to $\tfrac{3}{4}$ is $-\tfrac{4}{3}$, and to $-5$ it is $\tfrac{1}{5}$.
Write the point-slope equation of the line parallel to $y = 3x - 7$ that passes through the point $Q(-4, 1)$.
Write the point-slope equation of the line parallel to $y = 3x - 7$ that passes through the point $Q(-4, 1)$.
Find the point-slope equation of the line that is perpendicular to the line $y - 2 = \frac{3}{4}(x + 1)$ and passes through the origin.
Find the point-slope equation of the line that is perpendicular to the line $y - 2 = \frac{3}{4}(x + 1)$ and passes through the origin.
Obtain the point-slope equation of the line that passes through the point $P(-3, 5)$ and is parallel to the line joining the points $M(1, -2)$ and $N(4, 7)$.
Obtain the point-slope equation of the line that passes through the point $P(-3, 5)$ and is parallel to the line joining the points $M(1, -2)$ and $N(4, 7)$.
Linear Models: A Rate and a Point
In a real situation, the slope is the rate of change (dollars per hour, miles per hour…) and a known data point is the point $(x_1, y_1)$. That is exactly what point-slope form needs. You don't need the starting value.
When you get two data points instead of a rate, find the slope first. And when a problem measures time from a clock reading, it is often easiest to start your own clock at $0$.
A plumber charges $\$30$ for the service call fee and $\$20$ for each hour of work. If we know that for a job of $2$ hours they charged a total of $\$70…
A plumber charges $\$30$ for the service call fee and $\$20$ for each hour of work. If we know that for a job of $2$ hours they charged a total of $\$70$, write the point-slope equation that relates the total cost to the hours worked, using this known point as a reference.
A bicycle rental company charges a fixed amount for the service plus a constant price per hour of use. It is known that for a rental of $3$ hours, $\$14…
A bicycle rental company charges a fixed amount for the service plus a constant price per hour of use. It is known that for a rental of $3$ hours, $\$14$ is paid, and for a rental of $5$ hours, $\$20$ is paid. Determine the slope of the linear cost function and write its point-slope equation using the data from the first customer.
A car travels along a straight road at a constant speed. At $10:00$ hours it is at mile $45$ and at $12:30$ hours it passes mile $245$. Write the point-slope equation…
A car travels along a straight road at a constant speed. At $10:00$ hours it is at mile $45$ and at $12:30$ hours it passes mile $245$. Write the point-slope equation that models the position of the car as a function of the time elapsed since $10:00$ hours, expressing the time in hours.
More Models: Depreciation, Conversions, and Demand
Not every rate is positive. When a quantity falls steadily, the slope is negative. Sentences like "for every $\$5$ increase, sales drop by $10$" give you the slope directly: change in output over change in input.
Any two paired readings define a line. Temperature scales, for example, are linear: use two well-known pairs of readings (like the freezing and boiling points of water on each scale) to get the slope, then use one of them as your point.
The value of computer equipment depreciates linearly over time. If it was bought for $\$1200$ and after $4$ years its book value is $\$400$, write the point-slope…
The value of computer equipment depreciates linearly over time. If it was bought for $\$1200$ and after $4$ years its book value is $\$400$, write the point-slope equation that relates the value of the equipment to the years elapsed, using the point corresponding to the fourth year.
The Celsius ($^\circ C$) and Fahrenheit ($^\circ F$) temperature scales have a linear relationship. Knowing that water freezes at $0^\circ C$ ($32^\circ F$) and boils at…
The Celsius ($^\circ C$) and Fahrenheit ($^\circ F$) temperature scales have a linear relationship. Knowing that water freezes at $0^\circ C$ ($32^\circ F$) and boils at $100^\circ C$ ($212^\circ F$), deduce the point-slope equation that allows converting degrees Celsius to Fahrenheit using the freezing point. Then, use it to calculate the temperature in Fahrenheit equivalent to the normal human body temperature of $37^\circ C$.
In a simple economic model, the quantity demanded of a product decreases linearly as its price increases. At a price of $\$20$, $500…
In a simple economic model, the quantity demanded of a product decreases linearly as its price increases. At a price of $\$20$, $500$ units are demanded, and it has been observed that for every increase of $\$2$ in price, the demand falls by $50$ units. Write the point-slope equation of the demand using the first piece of data as a reference, and calculate at what price the demand will be zero.
Geometry with Lines
Point-slope form is the main tool for coordinate geometry.
An altitude of a triangle is the segment from a vertex perpendicular to the opposite side. To find its line, use the vertex as the point and the negative reciprocal of the opposite side's slope.
Check analytically if the points $A(-2, 4)$, $B(1, -2)$ and $C(5, -10)$ are collinear. To do this, first find the point-slope equation of the line that passes through…
Check analytically if the points $A(-2, 4)$, $B(1, -2)$ and $C(5, -10)$ are collinear. To do this, first find the point-slope equation of the line that passes through $A$ and $B$, and then verify if the coordinates of $C$ satisfy this equation.
Find the point-slope equation of the perpendicular bisector of the line segment with endpoints $A(-4, 2)$ and $B(2, 6)$. Remember that the perpendicular bisector is the…
Find the point-slope equation of the perpendicular bisector of the line segment with endpoints $A(-4, 2)$ and $B(2, 6)$. Remember that the perpendicular bisector is the line perpendicular to the segment that passes through its midpoint.
Given the triangle with vertices $A(1, 1)$, $B(7, 4)$ and $C(3, 7)$, find the point-slope equation of the altitude corresponding to the vertex $C$.
Given the triangle with vertices $A(1, 1)$, $B(7, 4)$ and $C(3, 7)$, find the point-slope equation of the altitude corresponding to the vertex $C$.
Intercepts, Intersections, and Reflections
Here are three tools for tougher problems.
Intercepts. To find where a line crosses the $x$-axis, set $y = 0$. To find where it crosses the $y$-axis, set $x = 0$. The axes and the line form a right triangle, with area $\tfrac{1}{2} \cdot (\text{base}) \cdot (\text{height})$, where the base and height are the lengths of the legs (always positive).
Intersections and unknown slopes. To find where two lines meet, solve the system. To make a line pass through a given point, substitute that point's coordinates into the equation and solve for the unknown.
Reflections. A mirror along the $x$-axis flips the direction of the ray vertically, so the slope changes sign. The ray hits the mirror at its $x$-intercept, and that is the point you use for the reflected line.
When the slope itself is unknown, write the intercepts in terms of $m$, use the area condition, and solve for $m$.
Determine the value of the parameter $k$ so that the line with equation $y - 3 = k(x + 2)$ passes through the point of intersection of the lines $r: 2x + y - 8 = 0$ and…
Determine the value of the parameter $k$ so that the line with equation $y - 3 = k(x + 2)$ passes through the point of intersection of the lines $r: 2x + y - 8 = 0$ and $s: x - 3y + 3 = 0$.
A ray of light follows a straight-line trajectory given by the equation $y - 5 = -\frac{1}{2}(x - 2)$. Upon reaching the x-axis (which acts as a perfect plane mirror),…
A ray of light follows a straight-line trajectory given by the equation $y - 5 = -\frac{1}{2}(x - 2)$. Upon reaching the x-axis (which acts as a perfect plane mirror), the ray is reflected. Knowing that the angle of incidence is equal to the angle of reflection, find the point-slope equation of the reflected ray using the point of impact on the mirror as a reference.
Write the point-slope equation of the line that passes through the point $P(2, -1)$ and which, when intersecting the coordinate semi-axes (positive x-axis and negative…
Write the point-slope equation of the line that passes through the point $P(2, -1)$ and which, when intersecting the coordinate semi-axes (positive x-axis and negative y-axis), forms a right triangle of area $4$ square units.
Summary
Back to the start: Emma's scooter fits $C - 9 = 3(h - 2)$, so 5 hours cost $C = 9 + 3(3) = \$18$. The starting fee never mattered.