## Tutorial: Linear functions and models

This tutorial: Part B: Finding the equation of a line

(This topic is also in Section 1.3 in

*Finite Mathematics and Applied Calculus*)

**Note**To use this tutorial, you should be familiar with the concepts of the

*slope*and $y$-

*intercept*of a straight line in the coordinate plane. See %%partAtut to review these concepts. In %%partAtut we studied the meaning of the slope $m$ and intercept $b$ in a linear function $f(x) = mx + b$, but we did not spend much time talking about how to

*obtain*a linear function in the first place. Of course, if you are

*given*$m$ and $b$, there is no more work to do; for instance, the linear function with $m = 3$ and $b = -4$ is $f(x) = 3x - 4$. However, it often happens that you are not given the slope and $y$-intercept directly.

Given the slope and a point

If you know the slope of a line (so you know how steep the line is) and the coordinates of a single point on that line, then you should know what line it is, as there can be only one line passing through that point with that particular slope: Start at the given point, go one unit to the right and $m$ units vertically (up if positive or down if negative) to get a second point. Connecting them gives the desired line.
**%%Q**#[OK, so we would know how to draw it. But how do we get its equation?][Bien, entonces sabríamos cómo dibujarlo. Pero ¿cómo obtenemos su ecuación?]#

**%%A**#[As follows][Como sigue]#: #[Let's call the given point $(x_1, y_1).$ Then , if $(x, y)$ is any (other) point on the line, the slope has to be][Llamemos al punto dado $(x_1, y_1).$ Entonces, si $(x, y)$ es cualquier (otro) punto de la recta, la pendiente tiene que ser]#

$y = y_1 + m(x-x_1).$ \t \gap[40] #[Traditional version of the point-slope formula][Versión tradicional de la fórmula punto-pendiente.]#

#[Distributing the $m$ gives][Distribuir la $m$ da]#
$y = mx + (y_1 - mx_1),$

#[so that the $y$-intercept is][por lo que la intersección en $y$ es]#
$b = (y_1 - mx_1),$

**Point-slope formula**The equation of the line through $(x_1, y_1)$ with slope $m$ is

$y = mx + b$ \gap[40] \t #[where][donde]#
\\ $b = y_1 - mx_1$. \t #[value of $b$][valor de $b$]#

**When to apply the point-slope formula**Apply the point-slope formula to find the equation of a line whenever you are given information about a point and the slope of the line. The formula does not apply if the line is vertical, as then its slope is undefined.

**Equation of a vertical line**If the line is vertical its slope is undefined, and it has equation $x = c$, a constant. So, the vertical line through the point $(p,q)$ has equation $x = p.$

**Examples**The line through $(2,-3)$ with slope $4$ has

$y = mx + b = 4x - 11$.

**Your turn**

Now try the exercises in Section 1.3 in

*Finite Mathematics and Applied Calculus*. or move ahead to the next tutorial by pressing "Next tutorial" on the sidebar.*March 2024*

Copyright © 2024 Stefan Waner and Steven R. Costenoble