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.

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The graph of a polynomial function is a smooth curve that may or may not change direction, depending on its degree. The quadratic, y = x2, is one of the two simplest polynomials.

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## The cubic polynomial graph

The cubic, y = x3 is another simple polynomial. Both the cubic and the quadratic go through the origin and the point (1, 1).

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## The graph of the line y = x

Just 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.

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## The absolute value function

The 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).

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## The reciprocal of x

The graphs of y = 1/x and y = 1/x2 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).

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## The reciprocal of x2

The graph of y = 1/x2 is symmetric with respect to the y-axis (it’s a mirror image on either side).

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## The graph of the square root

The 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.

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## The graph of the cube root

The 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).

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## The graph of the exponential function

The graph of the exponential function y = ex 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.

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## The graph of the logarithmic function

The graph of the logarithmic function y = ln x is the mirror image of its inverse function, y = ex, 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.