Question

Determine whether each graph represents a linear or nonlinear function. Use the
graph to estimate the x- and y-intercepts. Identify the type of symmetry in each graph.

Answer 1

Answer:

• x-intercepts: -3, -2, -1, 1, 2
• y-intercept: 12
• symmetry: none

Step-by-step explanation:

The x- and y-intercepts are the points where the graph crosses the x- and y-axes, respectively. If an odd-degree polynomial function has any symmetry, it will be symmetrical about the origin.

X-intercepts

The graph shows crossings of the x-axis at -3, -2, -1, 1, and 2. These are the x-intercepts. Expressed as coordinate pairs, each will have a y-coordinate of zero:

  (-3, 0), (-2, 0), (-1, 0), (1, 0), (2, 0)

Y-intercepts

If the function is not vertically scaled, the y-intercept of an odd-degree function will be the opposite of the product of the x-intercepts:

  -(-3)(-2)(-1)(1)(2) = 12

It will have an x-coordinate of zero:

  (0, 12) . . . y-intercept

A function can have at most 1 y-intercept.

Symmetry

There are two features of this graph that tell you it is an odd-degree function:

1. the end behavior is in opposite directions
2. there are an odd number of x-intercepts (x-axis crossings)

An odd-degree function will only be symmetrical about the origin. This function has more negative x-intercepts than positive ones, so is not symmetrical about the origin.

The graph has no symmetry.

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Additional comment

Turning points without an x-axis-crossing on either side, or axis "touches" (not crossing) signify complex or even-multiplicity roots. The function will only be of odd degree if there are an odd number of x-axis crossings.

The end behavior of an odd-degree function will be in opposite directions: the infinity will match the sign of x if the leading coefficient is positive, or be opposite to the sign of x if the leading coefficient is negative.