Section 3

Linear functions: slope, intercept, and graph

Polynomial functions of degree one, or simply the linear functions take the following general form:
$$f(x)=a+bx$$ The main characteristic of a linear function is its slope which shows the steepness and direction of the function. The slope of a linear function, with any two points \((x_1,y_1 ) \) and \((x_2,y_2 )\) satisfying the function is:
$$b={{\Delta y}\over \Delta x}={{(y_2-y_1)}\over (x_2-x_1 )}={{rise}\over run}$$ If a linear function is \(f(x)=a+bx\), the term b is the slope of the function and \((0,a)\) is the y intercept. For example, given a function \(f_1 (x)=2+2x\), the slope is \(2\) and the vertical intercept is \((0,2)\). Consider another function \(f_2 (x)=2-2x\) which has the slope \(-2\) and vertical intercept \((0,2)\). If the slope of a function is positive, for example \(f_1 (x)=2+2x\), the graph of the function is upward sloping. On the other hand, if the slope is a negative number, the graph of the function is downward sloping. For example, \(f_2 (x)=2-2x\) is a downward sloping function. Consider a third example, \(f_3 (x)=2-4x\) which is also a downward sloping function with the slope equals \(-4\). However, \(f_3 (x)\) is steeper than the other two functions \(f_1 (x)\) and \(f_2 (x)\). The bigger the absolute value of the slope of a function, the steeper is the line representing the function.

Consider drawing the graphs of the following equations. $$3y + 9x = 27 \quad (1)$$ $$3y - 9x = 27 \quad (2)$$

We can rewrite equation (1) as:

The graph representing equation (1) has the slope \(b=-3\) and vertical intercept \((0,9)\). The corresponding line is a downward sloping line. (See Figure 4)
Similarly, we can rewrite equation (2) as a function of x as follows:

The graph representing equation (2) has the slope \(b=3\) and vertical intercept \((0,9)\). The corresponding line is an upward sloping line (See Figure 4).