6.6: Analyzing the Graph of Polynomial Functions
Objective
To learn about the parts of a polynomial function and how to graph them. The graphing calculator will be used to aid in graphing.
Review Queue
1. Graph \begin{align*}4x-5y=25\end{align*}. Find the slope, \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}intercepts.
2. Graph \begin{align*}y=x^2-2x-8\end{align*}. Find the \begin{align*}x-\end{align*}intercepts, \begin{align*}y-\end{align*}intercept, and vertex.
3. Find the vertex of \begin{align*}y=-4x^2+24x+5\end{align*}. Is the vertex a maximum or minimum?
Finding and Defining Parts of a Polynomial Function Graph
Objective
Learning about the different parts of graphs for higher-degree polynomials.
Watch This
First watch this video.
James Sousa: Ex: Increasing/ Decreasing/ Relative Extrema from Analyzing a Graph
Then watch this video.
James Sousa: Summary of End Behavior or Long Run Behavior of Polynomial Functions
Guidance
By now, you should be familiar with the general idea of what a polynomial function graph does. It should cross the \begin{align*}x-\end{align*}axis as many times as the degree, unless there are imaginary solutions. It will curve up and down and can have a maximum and a minimum. Let’s define the parts of a polynomial function graph here.
Notice that in both the cubic (third degree, on the left) and the quartic (fourth degree, on the right) functions, there is no vertex. We now have minimums and maximums. If there are more than one minimum or maximum, there will be an absolute maximum/minimum, which is the lowest/highest point of the graph. A local maximum/minimum is a maximum/minimum relative to the points around it. The places where the function crosses the \begin{align*}x-\end{align*}axis are still the solutions (also called \begin{align*}x-\end{align*}intercepts, roots or zeros). In the quartic function, there is a repeated root at \begin{align*}x = 4\end{align*}. A repeated root will touch the \begin{align*}x-\end{align*}axis without passing through or it can also have a “jump” in the curve at that point (see Example A). All of these points together (maximums, minimums, \begin{align*}x-\end{align*}intercepts, and \begin{align*}y-\end{align*}intercept) are called critical values.
Another important thing to note is end behavior. It is exactly what it sounds like; how the “ends” of the graph behaves or points. The cubic function above has ends that point in the opposite direction. We say that from left to right, this function is mostly increasing. The quartic function’s ends point in the same direction, both positive, just like a quadratic function. When considering end behavior, look at the leading coefficient and the degree of the polynomial.
Example A
Use a table to graph \begin{align*}y=x^3\end{align*}.
Solution: Draw a table and pick at least 5 values for \begin{align*}x\end{align*}.
\begin{align*}x\end{align*} | \begin{align*}x^3\end{align*} | \begin{align*}y\end{align*} |
---|---|---|
-2 | \begin{align*}(-2)^3\end{align*} | -8 |
-1 | \begin{align*}(-1)^3\end{align*} | -1 |
0 | \begin{align*}0^3\end{align*} | 0 |
1 | \begin{align*}1^3\end{align*} | 1 |
2 | \begin{align*}2^3\end{align*} | 8 |
Plot the points and connect. This particular function is the parent graph for cubic functions. Recall from quadratic functions, that the parent graph has a leading coefficient of 1, no other \begin{align*}x-\end{align*}terms, and no \begin{align*}y-\end{align*}intercept. \begin{align*}y=x^4\end{align*} and \begin{align*}y=x^5\end{align*} are also parent graphs.
Example B
Analyze the graph below. Find the critical values, end behavior, and find the domain and range.
Solution: First, find the solutions. They appear to be (-2, 0), (1, 0), and (2, 0). Therefore, this function has a minimum degree of 3. However, look at the \begin{align*}y-\end{align*}intercept. The graph slightly bends between the maximum and minimum. This movement in the graph tells us that there are two imaginary solutions (recall that imaginary solutions always come in pairs). Therefore, the function has a degree of 5. Approximate the other critical values:
maximum: (-1.1, 10)
minimum: (1.5, -1.3)
\begin{align*}y-\end{align*}intercept: (0, 5)
In general, this function is mostly increasing and the ends go in opposite directions. The domain and range are both all real numbers.
When describing critical values, you may approximate their location. In the next concept, we will use the graphing calculator to find these values exactly.
Sometime it can be tricky to see if a function has imaginary solutions from the graph. Compare the graph in Example B to the cubic function above. Notice that it is smooth between the maximum and minimum. As was pointed out earlier, the graph from Example B bends. Any function with imaginary solutions will have a slightly irregular shape or bend like this one does.
Example C
Sketch a graph of a function with roots \begin{align*}-4, -3, \frac{1}{2}\end{align*}, and 3, has an absolute maximum at (2, 5), and has negative end behavior. This function does not have any imaginary roots.
Solution: There are several possible answers for this graph because we are only asking for a sketch. You would need more information to get an exact answer. Because this function has negative end behavior and four roots, we know that it will pass through the \begin{align*}x-\end{align*}axis four times and face down. The absolute maximum is located between the roots \begin{align*}\frac{1}{2}\end{align*} and 3. Plot these five points and connect to form a graph.
Guided Practice
1. Use a table to graph \begin{align*}f(x)=-(x+2)^2(x-3)\end{align*}.
2. Analyze the graph. Find all the critical values, domain, range and describe the end behavior.
3. Draw a graph of the cubic function with solutions of -6 and a repeated root at 1. This function is generally increasing and has a maximum value of 9.
Answers
1. This function is in intercept form. Because the factor, \begin{align*}(x + 2)\end{align*} is squared, we know it is a repeated root. Therefore, the function should just touch at -2 and not pass through the \begin{align*}x-\end{align*}axis. There is also a zero at 3. Because the function is negative, it will be generally decreasing. Think of the slope of the line between the two endpoints. It would be negative. Select several points around the zeros to see the behavior of the graph.
\begin{align*}x\end{align*} | \begin{align*}y\end{align*} |
---|---|
-4 | 14 |
-2 | 0 |
0 | 12 |
2 | 16 |
3 | 0 |
4 | -36 |
2. There are three real zeros at approximately -3.5, 1, and 7. Notice the curve between the zeros 1 and 7. This indicated there are two imaginary zeros, making this at least a fifth-degree polynomial. Think about an imaginary horizontal line at \begin{align*}y = 3\end{align*}. This line would touch the graph five times, so there should be five solutions. Next, there is an absolute minimum at (-0.5, -7.5), a local maximum at (2.25, 5), a local minimum at (2.25, 2.25) and an absolute maximum at (5, 6). The \begin{align*}y-\end{align*}intercept is at (0, -6). The domain and range are both all real numbers and the end behavior is mostly decreasing.
3. To say the function is “mostly increasing” means that the slope of the line that connects the two ends (arrows) is positive. Then, the function must pass through (-6, 0) and touch, but not pass through (1, 0). From this information, the maximum must occur between the two zeros and the minimum will be the double root.
Vocabulary
- Absolute Maximum/Minimum
- The highest/lowest point of a function. When referring to the absolute maximum/minimum value, use the \begin{align*}y-\end{align*}value.
- Local Maximum/Minimum
- The highest/lowest point relative to the points around it. A function can have multiple local maximums or minimums.
Solutions:
The \begin{align*}x-\end{align*}intercepts. Also called roots or zeros.
Critical Values: The \begin{align*}x-\end{align*}intercepts, maximums, minimums, and \begin{align*}y-\end{align*}intercept.
End Behavior: How the ends of a graph look. End behavior depends on the degree of the function and the leading coefficient.
Parent Graph: The most basic function of a particular type. It has a leading coefficient of 1, no additional \begin{align*}x-\end{align*}terms, and no constant.
Problem Set
Use the given \begin{align*}x-\end{align*}values to make a table and graph the functions below.
- \begin{align*}f(x) &= x^3-7x^2+15x-2\\ x &= -2, -1, 0, 1, 2, 3, 4\end{align*}
- \begin{align*}g(x) &= -2x^4 - 11x^3 - 3x^2+37x+35\\ x &= -5, -4, -3, -2, -1, 0, 1, 2\end{align*}
- \begin{align*}y &=2x^3+25x^2+100x+125\\ x &= -7,-6,-5,-4,-3,-2,-1,0\end{align*}
Make your own table and graph the following functions.
- \begin{align*}f(x)=(x+5)(x+2)(x-1)\end{align*}
- \begin{align*}y=x^4\end{align*}
- \begin{align*}y=x^5\end{align*}
- Analyze the graphs of \begin{align*}y=x^2, y=x^3, y=x^4\end{align*}, and \begin{align*}y=x^5\end{align*}. These are all parent functions. What do you think the graph of \begin{align*}y=x^6\end{align*} and \begin{align*}y=x^7\end{align*} will look like? What can you say about the end behavior of all even functions? Odd functions? What are the solutions to these functions?
- Writing How many repeated roots can one function have? Why?
Analyze the graphs of the following functions. Find all critical values, the domain, range, and end behavior.
For questions 13-15, make a sketch of the following real-solution functions.
- Draw two different graphs of a cubic function with zeros of -1, 1, and 4.5 and a minimum of -4.
- A fourth-degree polynomial with roots of -3.2, -0.9, 1.2, and 8.7, positive end behavior, and a local minimum of -1.7.
- A fourth-degree function with solutions of -7, -4, 1, and 2, negative end behavior, and an absolute maximum at \begin{align*}\left(-\frac{11}{2}, \frac{1755}{128}\right)\end{align*}.
- Challenge Find the equation of the function from #15.
Graphing Polynomial Functions with a Graphing Calculator
Objective
To graph polynomial functions and find critical values using a graphing calculator.
Watch This
James Sousa: Ex: Solve a Polynomial Equation Using a Graphing Calculator (Approximate Solutions)
Guidance
In the Quadratic Functions chapter, you used the graphing calculator to graph parabolas. Now, we will expand upon that knowledge and graph higher-degree polynomials. Then, we will use the graphing calculator to find the zeros, maximums and minimums.
Example A
Graph \begin{align*}f(x)=x^3+x^2-8x-8\end{align*} using a graphing calculator.
Solution: These instructions are for a TI-83 or 84. First, press \begin{align*}Y=\end{align*}. If there are any functions in this window, clear them out by highlighting the = sign and pressing ENTER. Now, in \begin{align*}Y1\end{align*}, enter in the polynomial. It should look like: \begin{align*}x^\land 3+x^\land 2-8x-8\end{align*}. Press GRAPH.
To adjust the window, press ZOOM. To get the typical -10 to 10 screen (for both axes), press 6:ZStandard. To zoom out, press ZOOM, 3:ZoomOut, ENTER, ENTER. For this particular function, the window needs to go from -15 to 15 for both \begin{align*}x\end{align*} and \begin{align*}y\end{align*}. To manually input the window, press WINDOW and change the \begin{align*}Xmin, Xmax, Ymin,\end{align*} and \begin{align*}Ymax\end{align*} so that you can see the zeros, minimum and maximum. Your graph should look like the one to the right.
Example B
Find the zeros, maximum, and minimum of the function from Example A.
Solution: To find the zeros, press \begin{align*}2^{nd}\end{align*} TRACE to get the CALC menu. Select 2:Zero and you will be asked “Left Bound?” by the calculator. Move the cursor (by pressing the \begin{align*}\uparrow\end{align*} or \begin{align*}\downarrow\end{align*}) so that it is just to the left of one zero. Press ENTER. Then, it will ask “Right Bound?” Move the cursor just to the right of that zero. Press ENTER. The calculator will then ask “Guess?” At this point, you can enter in what you think the zero is and press ENTER again. Then the calculator will give you the exact zero. For the graph from Example A, you will need to repeat this three times. The zeros are -2.83, -1, and 2.83.
To find the minimum and maximum, the process is almost identical to finding zeros. Instead of selecting 2:Zero, select 3:min or 4:max. The minimum is (1.33, -14.52) and the maximum is (-2, 4).
Example C
Find the \begin{align*}y-\end{align*}intercept of the graph from Example A.
Solution: If you decide not to use the calculator, plug in zero for \begin{align*}x\end{align*} and solve for \begin{align*}y\end{align*}.
\begin{align*}f(0) &= 0^3+0^2 - 8 \cdot 0 - 8\\ &= -8\end{align*}
Using the graphing calculator, press \begin{align*}2^{nd}\end{align*} TRACE to get the CALC menu. Select 1:value. \begin{align*}X=\end{align*} shows up at the bottom of the screen. If there is a value there, press CLEAR to remove it. Then press 0 and ENTER. The calculator should then say “\begin{align*}Y=-8\end{align*}.”
Guided Practice
Graph and find the critical values of the following functions.
1. \begin{align*}f(x)=-\frac{1}{3}x^4-x^3+10x^2+25x-4\end{align*}
2. \begin{align*}g(x)=2x^5-x^4+6x^3+18x^2-3x-8\end{align*}
3. Find the domain and range of the previous two functions.
4. Describe the types of solutions, as specifically as possible, for question 2.
Answers
Use the steps given in Examples \begin{align*}A, B\end{align*}, and \begin{align*}C\end{align*}.
1. zeros: -5.874, -2.56, 0.151, 5.283
\begin{align*}y-\end{align*}intercept: (0, -4)
minimum: (-1.15, -18.59)
local maximum: (-4.62, 40.69)
absolute maximum: (3.52, 113.12)
2. zeros: -1.413, -0.682, 0.672
\begin{align*}y-\end{align*}intercept: (0, -8)
minimum: (-1.11, 4.41)
maximum: (0.08, -8.12)
3. The domain of #1 is all real numbers and the range is all real numbers less than the maximum; \begin{align*}(-\infty, 113.12]\end{align*}. The domain and range of #2 are all real numbers.
4. There are three irrational solutions and two imaginary solutions.
Problem Set
Graph questions 1-6 on your graphing calculator. Sketch the graph in an appropriate window. Then, find all the critical values, domain, range, and describe the end behavior.
- \begin{align*}f(x)=2x^3+5x^2-4x-12\end{align*}
- \begin{align*}h(x)=-\frac{1}{4}x^4-2x^3-\frac{13}{4} x^2-8x-9\end{align*}
- \begin{align*}y=x^3-8\end{align*}
- \begin{align*}g(x)=-x^3-11x^2-14x+10\end{align*}
- \begin{align*}f(x)=2x^4+3x^3-26x^2-3x+54\end{align*}
- \begin{align*}y=x^4+2x^3-5x^2-12x-6\end{align*}
- What are the types of solutions in #2?
- Find the two imaginary solutions in #3.
- Find the exact values of the irrational roots in #5.
Determine if the following statements are SOMETIMES, ALWAYS, or NEVER true. Explain your reasoning.
- The range of an even function is \begin{align*}(-\infty, max]\end{align*}, where max is the maximum of the function.
- The domain and range of all odd functions are all real numbers.
- A function can have exactly three imaginary solutions.
- An \begin{align*}n^{th}\end{align*} degree polynomial has \begin{align*}n\end{align*} real zeros.
- Challenge The exact value for one of the zeros in #2 is \begin{align*}-4+\sqrt{7}\end{align*}. What is the exact value of the other root? Use this information to find the imaginary roots.
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