Slope and intercept

Linear formulas and equations: Linear functions

Slope and intercept

SlopeIn the linear formula of the form

$y=\blue a \cdot x + \green b$ with $\blue a$ and $\green b$ being numbers $\blue a$ is called the slope (or gradient). The slope indicates the direction of the line.

The slope can be calculated in a graph by dividing the vertical change ${\Delta y}$ by the horizontal change ${\Delta x}$ :

$\blue {a} =\frac{{\Delta y}}{{\Delta x}}$

If $\Delta x = 1$ , then $\Delta y = \blue a$ . In other words, if we move $1$ to the right in the graph, we increase by $\blue a$ . This can be seen in the graph on the right-hand side.

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Determining the slope from two points

Assume that the linear formula moves through the points $\blue{\rv{1, 3}}$ and $\green{\rv{3,7}}$ . Then, the slope will be equal to the vertical change divided by the horizontal change. On the horizontal axis, the $x$ -value changes from $1$ to $3$ , while on the vertical axis,s the $y$ -value changes from $3$ to $7$ . This gives:

$\frac{{\Delta y}}{{\Delta x}}=\frac{7-3}{3-1}=\frac{4}{2}=2$

In general, if a linear formula moves through the points $\blue A$ with coordinates $\blue{\rv{x_A, y_A}}$ and $\green{B}$ with coordinates $\green{\rv{x_B,y_B}}$ , then the slope is equal to:

$\frac{{\Delta y}}{{\Delta x}}=\frac{y_B-y_A}{x_B-x_A}$

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Intercept

In the linear formula of the form $y=\blue a \cdot x +\green b$ with $\blue a$ and $\green b$ numbers, the number $\green b$ is the $\green{\text{intercept}}$ .

The intercept indicates at which $y$ -value the graph of the formula intersects with the $y$ -axis. Hence, the point $\rv{0,\green b}$ lies on the graph of the formula.

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Recognizing the intercept in a table

Since the intercept indicates for which $y$ -value the formula intersects the $y$ -axis, the intercept is equal to the value of the formula at $x=0$ . Hence, in a table, we can read the intercept at $x=0$ .

Example

The following table belongs to a linear formula.

$x$	$-2$	$-1$	$0$	$1$	$2$
$y$	$-6$	$-8$	$-10$	$-12$	$-14$

The intercept can be found by looking at the value of $x=0$ . In this case, we have $y=-10$ . This means that the intercept of this formula is equal to $-10$ .

Determine the slope of the line given by the formula $y={{x}\over{21}}-{{1}\over{9}}$ .

The slope is: ${{1}\over{21}}$

The slope of the line from the formula $y=a\cdot x+b$ , in which $a$ and $b$ are numbers, is equal to $a$ .

The given formula $y={{x}\over{21}}-{{1}\over{9}}$ already has the form $y=a\cdot x+b$ , which means you can directly determine the slope: it is the coefficient of $x$ , which is equal to ${{1}\over{21}}$ .

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