# 10 Basic Algebraic Graphs

## The quadratic polynomial graph

The 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.

## The cubic polynomial graph

The cubic, *y* = *x*^{3} is another simple polynomial. Both the cubic and the quadratic go through the origin and the point (1, 1).

## 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.

## 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.

## 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).

## The reciprocal of *x*

The 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).

## 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).

## 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).

## 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.

## 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).

## The graph of the exponential function

The 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.

## The graph of the exponential function

The 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.

## The graph of the logarithmic function

The 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.