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Mathematics LibreTexts

3.3 Graphs of Polynomial Functions

In the previous section we explored the short run behavior of quadratics, a special case of polynomials. In this section we will explore the short run behavior of polynomials in general.

Short run Behavior: Intercepts

As with any function, the vertical intercept can be found by evaluating the function at an input of zero. Since this is evaluation, it is relatively easy to do it for a polynomial of any degree.

To find horizontal intercepts, we need to solve for when the output will be zero. For general polynomials, this can be a challenging prospect. While quadratics can be solved using the relatively simple quadratic formula, the corresponding formulas for cubic and 4th degree polynomials are not simple enough to remember, and formulas do not exist for general higher-degree polynomials.  Consequently, we will limit ourselves to three cases:

  1. The polynomial can be factored using known methods: greatest common factor and trinomial factoring. 
  2. The polynomial is given in factored form.
  3. Technology is used to determine the intercepts.

 

Example 1

Find the horizontal intercepts of File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image002.gif.

SOLUTION

We can attempt to factor this polynomial to find solutions for f(x) = 0.

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image004.gif               Factoring out the greatest common factor

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image006.gif             Factoring the inside as a quadratic in x2

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image008.gif           Then break apart to find solutions

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image010.gif          or         File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image012.gif     or         File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image014.gif

This gives us 5 horizontal intercepts.

Example 2

Find the vertical and horizontal intercepts of File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image016.gif

SOLUTION

The vertical intercept can be found by evaluating \(g(0)\). 

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image018.gif

The horizontal intercepts can be found by solving \(g(t) = 0\)

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image020.gif               Since this is already factored, we can break it apart:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image022.gif  or         File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image024.gif

We can always check our answers are reasonable by graphing the polynomial.

Description: Description: Graphs

Example 3

Find the horizontal intercepts of File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image028.gif

SOLUTION

Since this polynomial is not in factored form, has no common factors, and does not appear to be factorable using techniques we know, we can turn to technology to find the intercepts. 

Graphing this function, it appears there are horizontal intercepts at t = -3, -2, and 1.

We could check these are correct by plugging in these values for t and verifying that File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image030.gif.

Try it Now

1. Find the vertical and horizontal intercepts of the functionFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image032.gif.

Graphical Behavior at Intercepts

Description: Description: Graphs

If we graph the function File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image036.gif, notice that the behavior at each of the horizontal intercepts is different.

At the horizontal intercept x = -3, coming from the File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image038.gif factor of the polynomial, the graph passes directly through the horizontal intercept.  The factor is linear (has a power of 1), so the behavior near the intercept is like that of a line - it passes directly through the intercept. We call this a single zero, since the zero corresponds to a single factor of the function.

At the horizontal intercept x = 2, coming from the File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image040.gif factor of the polynomial, the graph touches the axis at the intercept and changes direction.  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.  Since File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image042.gif, the factor is repeated twice, so we call this a double zero.

At the horizontal intercept x = -1, coming from the File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image044.gif factor of the polynomial, the graph passes through the axis at the intercept, but flattens out a bit first.  This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic, with the same “S” type shape near the intercept that the toolkit File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image046.gif has. We call this a triple zero.

By utilizing these behaviors, we can sketch a reasonable graph of a factored polynomial function without needing technology.

Graphical Behavior of Polynomials at Horizontal Intercepts

If a polynomial contains a factor of the form File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image048.gif, the behavior near the horizontal intercept h is determined by the power on the factor.

          p = 1                                        p = 2                                        p = 3

Description: Description: Graphs        Description: Description: Graphs         Description: Description: Graphs

          Single zero                      Double zero                                Triple zero

 

For higher even powers 4,6,8 etc.… the graph will still bounce off of the horizontal axis but the graph will appear flatter with each increasing even power as it approaches and leaves the axis.

For higher odd powers, 5,7,9 etc… the graph will still pass through the horizontal axis but the graph will appear flatter with each increasing odd power as it approaches and leaves the axis.

Example 4

Sketch a graph of File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image056.gif.

SOLUTION

This graph has two horizontal intercepts. At \(x = -3\), the factor is squared, indicating the graph will bounce at this horizontal intercept.  At x = 5, the factor is not squared, indicating the graph will pass through the axis at this intercept.

Additionally, we can see the leading term, if this polynomial were multiplied out, would be File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image058.gif, so the long-run behavior is that of a vertically reflected cubic, with the outputs decreasing as the inputs get large positive, and the inputs increasing as the inputs get large negative.

To sketch this we consider the following:

As File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image060.gif the function File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image062.gif so we know the graph starts in the 2nd quadrant and is decreasing toward the horizontal axis.

At (-3, 0) the graph bounces off of the horizontal axis and so the function must start increasing.

At (0, 90) the graph crosses the vertical axis at the vertical intercept.

Somewhere after this point, the graph must turn back down or start decreasing toward the horizontal axis since the graph passes through the next intercept at (5,0).

Description: Description: Graphs

As File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image066.gif the functionFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image068.gif so we know the graph continues to decrease and we can stop drawing the graph in the 4th quadrant.

 

Using technology we can verify that the resulting graph will look like:

Solving Polynomial Inequalities

One application of our ability to find intercepts and sketch a graph of polynomials is the ability to solve polynomial inequalities. It is a very common question to ask when a function will be positive and negative. We can solve polynomial inequalities by either utilizing the graph, or by using test values.

Example 5

Solve File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image070.gif

SOLUTION

As with all inequalities, we start by solving the equality File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image072.gif, which has solutions at x = -3, -1, and 4.  We know the function can only change from positive to negative at these values, so these divide the inputs into 4 intervals. 

We could choose a test value in each interval and evaluate the function File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image074.gif at each test value to determine if the function is positive or negative in that interval

Text Box: Interval	Test x in interval	f( test value)	>0 or <0?
x < -3	-4	72	> 0
-3 < x < -1	-2	-6	< 0
-1 < x < 4	0	-12	< 0
x > 4	5	288	> 0

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image076.gifOn a number line this would look like:

 

Description: Description: Graphs

From our test values, we can determine this function is positive when x < -3 or x > 4, or in interval notation, File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image080.gif

We could have also determined on which intervals the function was positive by sketching a graph of the function.  We illustrate that technique in the next example

Example 6

Find the domain of the function File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image082.gif.

SOLUTION

A square root is only defined when the quantity we are taking the square root of, the quantity inside the square root, is zero or greater.  Thus, the domain of this function will be when File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image084.gif.

 

Description: Description: GraphsAgain we start by solving the equality File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image088.gif.  While we could use the quadratic formula, this equation factors nicely to File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image090.gif, giving horizontal intercepts t = 1 and t = -6.  Sketching a graph of this quadratic will allow us to determine when it is positive.

 

 

From the graph we can see this function is positive for inputs between the intercepts.  So File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image084.gif for File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image092.gif, and this will be the domain of the v(t) function.

Try it Now

2. Given the function File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image094.gif use the methods that we have learned so far to find the vertical & horizontal intercepts, determine where the function is negative and positive, describe the long run behavior and sketch the graph without technology.

Writing Equations using Intercepts

Since a polynomial function written in factored form will have a horizontal intercept where each factor is equal to zero, we can form a function that will pass through a set of horizontal intercepts by introducing a corresponding set of factors.

 

Factored Form of Polynomials

If a polynomial has horizontal intercepts at File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image096.gif, then the polynomial can be written in the factored form

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image098.gif

where the powers pi on each factor can be determined by the behavior of the graph at the corresponding intercept, and the stretch factor a can be determined given a value of the function other than the horizontal intercept.

Example 7

Write a formula for the polynomial function graphed here.

Description: Description: Graphs

SOLUTION

 

This graph has three horizontal intercepts: x = -3, 2, and 5.  At x = -3 and 5 the graph passes through the axis, suggesting the corresponding factors of the polynomial will be linear.  At x = 2 the graph bounces at the intercept, suggesting the corresponding factor of the polynomial will be 2nd degree (quadratic).  Together, this gives us:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image102.gif

To determine the stretch factor, we can utilize another point on the graph.  Here, the vertical intercept appears to be (0,-2), so we can plug in those values to solve for a:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image104.gif

The graphed polynomial appears to represent the function File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image106.gif.

Try it Now

3. Given the graph, write a formula for the function shown.

 

Description: Description: Graphs

Estimating Extrema

With quadratics, we were able to algebraically find the maximum or minimum value of the function by finding the vertex. For general polynomials, finding these turning points is not possible without more advanced techniques from calculus. Even then, finding where extrema occur can still be algebraically challenging. For now, we will estimate the locations of turning points using technology to generate a graph.

Example 8
An open-top box is to be constructed by cutting out squares from each corner of a 14 cm by 20 cm sheet of plastic then folding up the sides.  Find the size of squares that should be cut out to maximize the volume enclosed by the box.

SOLUTION

We will start this problem by drawing a picture, labeling the width of the cut-out squares with a variable, w.     

Notice that after a square is cut out from each end, it leaves a (14-2w) cm by (20-2w) cm rectangle for the base of the box, and the box will be w cm tall.  This gives the volume:

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image111.gif

Using technology to sketch a graph allows us to estimate the maximum value for the volume, restricted to reasonable values for w: values from 0 to 7.

Description: Description: Graphs

 

From this graph, we can estimate the maximum value is around 340, and occurs when the squares are about 2.75cm square.  To improve this estimate, we could use advanced features of our technology, if available, or simply change our window to zoom in on our graph.

 

Description: Description: Graphs

From this zoomed-in view, we can refine our estimate for the max volume to about 339, when the squares are 2.7cm square.

File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image109.gif

 

 

Try it Now

4. Use technology to find the maximum and minimum values on the interval [-1, 4] of the functionFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image117.gif.

Important Topics of this Section

  • Short Run Behavior
    • Intercepts (Horizontal & Vertical)
  • Methods to find Horizontal intercepts
    • Factoring Methods
    • Factored Forms
    • Technology
  • Graphical Behavior at intercepts
  • Single, Double and Triple zeros (or power 1, 2, and 3 behaviors)
  • Solving polynomial inequalities using test values & graphing techniques
  • Writing equations using intercepts
  • Estimating extrema

 

Try it Now Answers

1.  Vertical intercept (0, 0), Horizontal intercepts (0, 0), (-2, 0), (2, 0)

2. Vertical intercept (0, 0), Horizontal intercepts (-2, 0), (0, 0), (3, 0)

    The function is negative on (File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image119.gif, -2) and (0, 3)

    The function is positive on (-2, 0) and (3,File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image121.gif)

    The leading term is File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image046.gifso asFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image060.gif, File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image125.gifand asFile:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image066.gif, File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image128.gif

 

Description: Description: Graphs

3. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image132.gif

4.  The minimum occurs at approximately the point (0, -6.5), and the maximum occurs at approximately the point (3.5, 7).

Section 3.3 Exercises

Find the C and t intercepts of each function.

1. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image134.gif                       2. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image136.gif

3. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image138.gif                               4. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image140.gif

5. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image142.gif                                  6. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image144.gif

Use your calculator or other graphing technology to solve graphically for the zeros of the function.

7. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image146.gif                          8. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image148.gif

Find the long run behavior of each function as File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image150.gif and File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image152.gif

9. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image154.gif                     10. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image156.gif

11. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image158.gif                           12. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image160.gif

 

Sketch a graph of each equation.

13. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image162.gif                             14. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image164.gif

15. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image166.gif                             16. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image168.gif

17. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image170.gif                         18. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image172.gif  

 

Solve each inequality.

19. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image174.gif                                    20. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image176.gif

21. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image178.gif                           22. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image180.gif  

 

Find the domain of each function.

23. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image182.gif                         24. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image184.gif

25. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image186.gif                                 26. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image188.gif

27. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image190.gif                           28. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image192.gif

29. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image194.gif                                      30. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image196.gif

Write an equation for a polynomial the given features.

  1. Degree 3.  Zeros at x = -2, x = 1, and x = 3.  Vertical intercept at (0, -4)
  2. Degree 3.  Zeros at x = -5, x = -2, and x = 1.  Vertical intercept at (0, 6)
  3. Degree 5.  Roots of multiplicity 2 at x = 3 and x = 1, and a root of multiplicity 1 at     x = -3.  Vertical intercept at (0, 9)
  4. Degree 4.  Root of multiplicity 2 at x = 4, and a roots of multiplicity 1 at x = 1 and     x = -2.  Vertical intercept at (0, -3)
  5. Degree 5.  Double zero at x = 1, and triple zero at x = 3.  Passes through the point    (2, 15)
  6. Degree 5.  Single zero at x = -2 and x = 3, and triple zero at x = 1.  Passes through the point (2, 4)

 

Write a formula for each polynomial function graphed.

 

37.Description: Description: http://localhost/imathas/filter/graph/svgimg.php?sscr=-5%2C5%2C-5%2C5%2C1%2C1%2C1%2Cnull%2Cnull%2C200%2C200%2Cfunc%2C1%2F2*%28x-1%29%28x%2B2%29%28x-3%29%2Cnull%2C0%2C0%2C%2C%2Cblack%2C1%2Cnone 38.Description: Description: http://localhost/imathas/filter/graph/svgimg.php?sscr=-5%2C5%2C-5%2C5%2C1%2C1%2C1%2Cnull%2Cnull%2C200%2C200%2Cfunc%2C-2%2F9%28x%2B3%29%28x%2B1%29%28x-3%29%2Cnull%2C0%2C0%2C%2C%2Cblack%2C1%2Cnone 39.Description: Description: http://localhost/imathas/filter/graph/svgimg.php?sscr=-5%2C5%2C-5%2C5%2C1%2C1%2C1%2Cnull%2Cnull%2C200%2C200%2Cfunc%2C-%28x%2B1%29%5E2%28x-2%29%2Cnull%2C0%2C0%2C%2C%2Cblack%2C1%2Cnone

40.Description: Description: http://localhost/imathas/filter/graph/svgimg.php?sscr=-5%2C5%2C-5%2C5%2C1%2C1%2C1%2Cnull%2Cnull%2C200%2C200%2Cfunc%2C1%2F4%28x%2B2%29%5E2%28x-3%29%2Cnull%2C0%2C0%2C%2C%2Cblack%2C1%2Cnone41.File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image206.jpg  42.File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image208.jpg

 

43.File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image210.jpg 44.File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image212.jpg

 

Write a formula for each polynomial function graphed.

 

45.File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image214.jpg                         46.File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image216.jpg

 

47.File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image218.jpg    48. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image220.jpg

 

49.File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image222.jpg    50. File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image224.jpg

 

  1. A rectangle is inscribed with its base on the x axis and its upper corners on the parabola File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image226.gif.  What are the dimensions of such a rectangle that has the greatest possible area?
  2. A rectangle is inscribed with its base on the x axis and its upper corners on the curve File:/C:\Users\DELMAR~1\AppData\Local\Temp\msohtmlclip1\01\clip_image228.gif.  What are the dimensions of such a rectangle that has the greatest possible area?

Contributors

  • David Lippman (Pierce College)
  • Melonie Rasmussen (Pierce College)