Sections 2.6 - Graphs of Special Functions

Section Objectives

1. Sketch the graph of $f(x)=ax+b$.
2. Sketch the graph of $f(x) = ax^2+c$.

In this section, we return briefly to graphing lines and parabolas.

Lines

The simplest way to graph a linear function of the form $f(x)=ax+b$ is to recognize that the function form is a slope-intercept form.

The graph of $f(x)=ax+b$ is a line with slope $a$ and $y$-intercept $(0,b)$. To graph the line, plot the point $(0,b)$ and use the slope to obtain another point on the line.

Examples

• Sketch the graph of $\displaystyle f(x)=\frac{2}{3} x -2$.

• Sketch the graph of $g(x)=-2x+4$.

Parabolas

The graph of $f(x)=ax^2+c$ is a parabola with vertex at $(0,c)$. If $a$ is positive, the graph opens upward. If $a$ is negative, the graph opens downward. In either case, the $y$-axis is a symmetry axis.

When graphing parabolas, you should plot a few points (besides the vertex) to pinpoint the precise shape.

Examples

• Describe the graph of $g(x)=x^2-6$.

• Sketch the graph of $\displaystyle f(x)=-\frac{1}{2}x^2+5$.