10 Basic Algebraic Graphs
Graphing is one way of getting the characteristics of a function out there for everyone to see. The basic graphs are just that — basic. They’re centered at the origin and aren’t expanded or shrunken or jostled about. You can alter the basic graphs by performing translations to the left or right or up or down.
1 The quadratic polynomial graphThe graph of a polynomial function is a smooth curve that may or may not change direction, depending on its degree. The quadratic, y = x^{2}, is one of the two simplest polynomials. |
2 The cubic polynomial graphThe cubic, y = x^{3} is another simple polynomial. Both the cubic and the quadratic go through the origin and the point (1, 1). |
3 The graph of the line y = xJust two points determine a unique line. This statement means that only one line can go through any two designated points. Lines can have x- and y-intercepts — where the lines cross the axes; the slope of a line tells whether it rises or falls and how steeply this happens. As the figure shows, the graph of the line y = x goes diagonally through the first and third quadrants. The slope is 1, and the line goes through the point (1, 1). The only intercept of this line is the origin. |
4 The absolute value functionThe absolute value function y = |x| has a characteristic V shape. The V is typical of most absolute value equations with linear terms. The only intercept of this basic absolute value graph is the origin, and the function goes through the point (1, 1). |
5 The reciprocal of xThe graphs of y = 1/x and y = 1/x^{2} both have vertical asymptotes of x = 0 and horizontal asymptotes of y = 0. The asymptotes are actually the x- and y-axes. Each curve goes through the point (1, 1), and each curve exhibits symmetry. The graph of y = 1/x is symmetric with respect to the origin (a 180-degree turn gives you the same graph). |
6 The reciprocal of x^{2}The graph of y = 1/x^{2} is symmetric with respect to the y-axis (it’s a mirror image on either side). |
7 The graph of the square rootThe graph of y = the square root of x starts at the origin and stays in the first quadrant. Except for (0, 0), all the points have positive x- and y-coordinates. The curve rises gently from left to right. |
8 The graph of the cube rootThe graph of y = the cube root of x is an odd function: It resembles, somewhat, twice its partner, the square root, with the square root curve spun around the origin into the third quadrant and made a bit steeper. You can take cube roots of negative numbers, so you can find negative x- and y- values for points on this curve. Both curves go through the point (1, 1). |
9 The graph of the exponential functionThe graph of the exponential function y = e^{x} is always above the x-axis. The only intercept of this graph is the y-intercept at (0, 1). The x-axis is the horizontal asymptote when x is very small, and the curve grows without bound as the x-values move to the right. |
10 The graph of the logarithmic functionThe graph of the logarithmic function y = ln x is the mirror image of its inverse function, y = e^{x}, over the line y = x. The function has one intercept, at (1, 0). The graph rises from left to right, moving from the fourth quadrant up through the first quadrant. The y-axis is the vertical asymptote as the values of x approach 0 — get very small. |