Reading: Graphs of Polynomial Functions (part II)
Graphical Behavior at Intercepts
Graphical Behavior of Polynomials at Horizontal Intercepts
If a polynomial contains a factor of the form (x – h)p, the behavior near the horizontal intercept h is determined by the power on the factor.Example 1
Sketch a graph of f(x) = –2(x + 3)2(x – 5). 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 –2x3, 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 x → –∞ the function f(x) → ∞ 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). As x → ∞ the function f(x) → –∞ 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 2
Solve (x + 3)(x + 1)2(x – 4) > 0 As with all inequalities, we start by solving the equality (x + 3)(x + 1)2(x – 4) = 0, 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 at each test value to determine if the function is positive or negative in that intervalInterval | 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 |
Example 3
Find the domain of the function [latex]\displaystyle{v}{({t})}=\sqrt{{{6}-{5}{t}-{t}^{{2}}}}[/latex] . 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 6 – 5 t – t2 $latex \ge$ 0.Try it Now 1
Given the function use the methods that we have learned so far to find the vertical and 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 x = x1,x2,...,xn, then the polynomial can be written in the factored form f(x) = a(x – x1)pn(x – x2)pn...(x – xn)pn 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 4
Write a formula for the polynomial function graphed here.Try it Now 2
Given the graph, write a formula for the function shown.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 5
An open-top box is to be constructed by cutting out squares from each corner of a 14cm by 20cm 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.Try it Now 3
Use technology to find the maximum and minimum values on the interval [–1, 4] of the function [latex]\displaystyle{f{{({x})}}}=-{0.2}{({x}-{2})}^{{3}}{({x}+{1})}^{{2}}{({x}-{4})}[/latex].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
- The function is negative on (–∞, –2) and (0, 3)The function is positive on (–2, 0) and (3,∞)The leading term is x3 so as x → –∞, g(x) → –∞ and as x → ∞, g(x) → ∞
- [latex]\displaystyle{f{{({x})}}}=-\frac{{1}}{{8}}{({x}-{2})}^{{3}}{({x}+{1})}^{{2}}{({x}-{4})}[/latex]
- The minimum occurs at approximately the point (0, –6.5), and the maximum occurs at approximately the point (3.5, 7).
David Lippman and Melonie Rasmussen, Open Text Bookstore, Precalculus: An Investigation of Functions, "Chapter 3: Polynomial and Rational Functions," licensed under a CC BY-SA 3.0 license.