# How to write a polynomial function with given zeros and y intercept

### How to write a polynomial function with given zeros and y intercept

The graph touches the x-axis, so the multiplicity of the zero must be even. Instructions: On the interactive graph there are two red dots. Technology is used to determine the intercepts. We can estimate the maximum value to be around cubic cm, which occurs when the squares are about 2. Look for the x-intercept where the graph crosses the x-axis. In these cases, we say that the turning point is a global maximum or a global minimum. Green check marks will appear if they are correct. Go to the applet at the bottom of the page. Putting this information together with the Leading Coefficient Test we can determine the end behavior of the graph of our given polynomial: Since the degree of the polynomial, 3, is odd and the leading coefficient, 5, is positive, then the graph of the given polynomial falls to the left and rises to the right. Hint Re-graph the points given, and continue making points in the pattern of the slope. Polynomial functions also display graphs that have no breaks. Do all polynomial functions have as their domain all real numbers? Find the size of squares that should be cut out to maximize the volume enclosed by the box. Interpreting Turning Points A turning point is a point of the graph where the graph changes from increasing to decreasing rising to falling or decreasing to increasing falling to rising.

Hints Remind yourself that a coordinate is x, y. In this section we will explore the local behavior of polynomials in general. The graph touches the x-axis, so the multiplicity of the zero must be even. If the graph touches the x-axis and bounces off of the axis, it is a zero with even multiplicity.

## How to write a polynomial function given a graph

For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the x-axis. If n is even AND the leading coefficient , is negative, the graph falls to the left and to the right. Over which intervals is the revenue for the company increasing? Look for the zeros of the linear function where the y-value is zero. Exercise 1 Which coordinate points represent the x- and y-intercepts of the graph shown below? For now, we will estimate the locations of turning points using technology to generate a graph. Even then, finding where extrema occur can still be algebraically challenging. This is because for very large inputs, say or 1,, the leading term dominates the size of the output. Find the size of squares that should be cut out to maximize the volume enclosed by the box. Example 3: Use the Leading Coefficient Test to determine the end behavior of the graph of the polynomial. Click "Get New Example. Now that we know how to find zeros of polynomial functions, we can use them to write formulas based on graphs. Other times, the graph will touch the horizontal axis and bounce off. The factor is quadratic degree 2 , so the behavior near the intercept is like that of a quadratic—it bounces off of the horizontal axis at the intercept.

If you said 5, you are right on!! Provided by: Lumen Learning. If you said 5, pat yourself on the back!! If the graph crosses the x-axis and appears almost linear at the intercept, it is a single zero. Did you notice something unusual about steps 2 and 3? This is probably a single zero of multiplicity 1. Click "Check" to see if you are correct. For zeros with odd multiplicities, the graphs cross or intersect the x-axis.

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Zeros and multiplicity