#[Visualizing the slope of a line][Visualiza la pendiente de una recta]#
Consider, for example, the linear function $f(x) = mx + b = 2x-1$ whose graph is shown below.
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$\color{darkred}{\dfrac{\Delta y}{\Delta x} = \dfrac{6}{3} = 2}$
$\color{darkgreen}{\dfrac{\Delta y}{\Delta x} = \dfrac{4}{2} = 2}$
$\color{blue}{\dfrac{\Delta y}{\Delta x} = \dfrac{2}{1} = 2}$
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#[ As we move along its graph, $y$ changes by $\Delta y = m = 2$ units for ever $1$-unit change $\Delta x$ in $x.$ In other words, the ratio $\dfrac{\Delta y}{\Delta x}$ is always the same, and equal to $m = 2.$ In general][ A medida que avanzamos en la gráfica, $y$ cambia en $\Delta y = m = 2$ unidades por cada cambio de $1$ unidad $\Delta x$ en $x$. En otras palabras, la relación $\dfrac{\Delta y}{\Delta x}$ siempre es lo mismo, e igual a $m = 2.$ En general,]#
#[Slope of a line][Pendiente de una recta]# $m = \dfrac{\Delta y}{\Delta x} = \dfrac{y_2-y_1}{x_2-x_1}$.
From the formula we see that the slope is positive when $\Delta x$ and $\Delta y$ have the same sign, and negative when they have opposite signs. The quotient $\Delta x/\Delta y$ defining the slope, being a quotient of two differences, is often called a
difference quotient.
#[Examples][Ejemplos]#
1. The line passing through $(-3,1)$ and $(5,2)$ has slope
$m = \dfrac{y_2-y_1}{x_2-x_1}$ \t ${}=\dfrac{2-1}{5-(-3)} = \dfrac{1}{8}.$
2. Below are two lines; one with a positive slope ($\Delta x$ and $\Delta y$ have the same sign) and one with a negative slope ($\Delta x$ and $\Delta y$ have opposite sign):
$(x_1,y_1)=(1,1)$, $(x_2,y_2)=(5,7)$
$m = \dfrac{\Delta y}{\Delta x} = \dfrac{7-1}{5-1} = \dfrac{6}{4}= \dfrac{3}{2}$
\t \t
$(x_1,y_1)=(1,7)$, $(x_2,y_2)=(5,1)$
$m = \dfrac{\Delta y}{\Delta x} = \dfrac{1-7}{5-1} = \dfrac{-6}{4}= -\dfrac{3}{2}$
#[$\Delta y$ is sometimes referred to as the "rise" (the amount the line goes up from left to right) and $\Delta x$ is referred to as the "run." In both graphs, the run is $\Delta x=4,$ but the rise $\Delta y$ is negative in the second graph. Notice also that switching the numbering of the two points results in the same quotient in either calculation (as both the numerator and denominator would change sign).][$\Delta y$ a veces se denomina "subida" (la cantidad que la línea sube de izquierda a derecha) y $\Delta x$ se denomina "corrida". En ambas gráficas, la corrida es $\Delta x = 4,$ pero la subida $\Delta y$ es negativa en la segunda gr´fica. Observa también que intercambiar la numeración de los dos puntos da como resultado el mismo cociente en cualquier cálculo (ya que tanto el numerador como el denominador cambiarían de signo).]#