Tutorial: Linear functions and models

Adaptive game version

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)

What is an adaptive game tutorial? ▶

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.