# How to find the turning point of a polynomial function

## 3.4: Graphs of Polynomial Functions A polynomial is an expression that deals with decreasing powers of 'x', such as in this example: 2X^3 + 3X^2 - X + 6. When a polynomial of degree two or higher.

Learning Objectives After completing this tutorial, you should be able to: Identify a polynomial function. Use the Leading Coefficient Test to find the end behavior of the graph of a given polynomial function. Find the zeros of a polynomial function. Find the multiplicity of a zero and know if the graph crosses the x-axis at the zero or touches the x-axis and turns around at the zero. Know the maximum number of turning points a graph of a polynomial function could have. Graph a polynomial function.

Graphs behave differently at various x -intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off. Figure 7. Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero. The factor is linear has a degree of 1 , so the behavior near the intercept is like that of a line—it passes directly through the intercept.

In particular, we are interested in locations where graph behavior changes. A turning point is a point at which the function values change from increasing to decreasing or decreasing to increasing. We are also interested in the intercepts. As with all functions, the y- intercept is the point at which the graph intersects the vertical axis. The point corresponds to the coordinate pair in which the input value is zero. The x- intercepts occur at the input values that correspond to an output value of zero. It is possible to have more than one x- intercept.

When a polynomial of degree two or higher is graphed, it produces a curve. This curve may change direction, where it starts off as a rising curve, then reaches a high point where it changes direction and becomes a downward curve. Conversely, the curve may decrease to a low point at which point it reverses direction and becomes a rising curve. If the degree is high enough, there may be several of these turning points. There can be as many turning points as one less than the degree -- the size of the largest exponent -- of the polynomial. Find the derivative of the polynomial.

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## TURNING POINTS OF POLYNOMIALS

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1. Caitlin L. says: