Tutorial: Linear functions and models

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

Go to Part C: Applications: Linear models

(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]#

$\dfrac{1}{1}$$m$ \t ${}= \dfrac{y - y_1}{x - x_1},$ \gap[30] #[so][por lo que]# \\ $y - y_1$ \t ${}=m(x-x_1).$

#[Solving for $y$ gives][Despejar a $y$ nos da]#

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