A polynomial is an expression that is the sum or difference of terms. It has real coefficients and integer exponents \(\ge0\).
- Standard Form: \(f\left(x\right)=ax^n+bx^{n-1}+cx^{n-2}+....+k\), where \(k\) is a constant
- Degree: The value of the largest exponent when a function is written in standard form (or the sum of the exponents when the function is written in factored form).
- Leading Coefficient: The number written in front of the variable with the largest exponent (or if written in factored form, the sum of the coefficients of \(x\) raised to the power of the factor).
The degree of a polynomial function and the sign of the leading coefficient can provide clues that let us quickly sketch a polynomial. Explore the graphs below and answer the questions that accompany them.
Graphs of odd degree polynomials
Quick Check 1
Quick Check 1
1) When the leading coefficient is positive
2) When the leading coefficient is negative
3) At even powered factors, what does the graph look like at the x-intercept? 4) At odd powered factors, what does the graph look like at the y-intercept? 5) How are the degree of the functions and the number of turns the graph makes related? Quick Check Solutions |
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Graphs of even degree polynomials
Quick Check 2
Quick Check 2
1) When the leading coefficient is positive
2) When the leading coefficient is negative
3) At even powered factors, what does the graph look like at the x-intercept? 4) At odd powered factors, what does the graph look like at the y-intercept? 5) How are the degree of the functions and the number of turns the graph makes related? Quick Check Solutions |
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The end behavior of a polynomial is the behavior of the \(y\) values (outputs) as the \(x\) values (inputs) approach positive or negative infinity.
The easiest way to remember what happens to \(y\) as \(x\) approaches positive infinity is to look at the sign of the leading coefficient. If the leading coefficient is positive, the \(y\) values go in an upward direction (towards positive infinity) and if the leading coefficient is negative, the \(y\) values go in a downward direction (towards negative infinity).
One other piece of information that is helpful when trying to determine the degree of a polynomial from its graph is to look at the number of extreme values (maxima and minima). The SMALLEST degree a polynomial will have is the number of extreme values plus one. For example, a graph that has a total of \(5\) extreme values will have a minimum degree of \(6\).
Example 1: Determine the least degree of the polynomial shown and the sign of the leading coefficient.
- If the degree of the polynomial is ODD, the \(y\) values will go in opposite directions. In other words, one side of the graph will approach positive infinity and one side of the graph will approach negative infinity.
- If the degree of the polynomial is EVEN, the \(y\) values will go in the same direction. In other words, both sides of the graph will approach positive infinity or negative infinity.
The easiest way to remember what happens to \(y\) as \(x\) approaches positive infinity is to look at the sign of the leading coefficient. If the leading coefficient is positive, the \(y\) values go in an upward direction (towards positive infinity) and if the leading coefficient is negative, the \(y\) values go in a downward direction (towards negative infinity).
One other piece of information that is helpful when trying to determine the degree of a polynomial from its graph is to look at the number of extreme values (maxima and minima). The SMALLEST degree a polynomial will have is the number of extreme values plus one. For example, a graph that has a total of \(5\) extreme values will have a minimum degree of \(6\).
Example 1: Determine the least degree of the polynomial shown and the sign of the leading coefficient.
The minimum degree of this polynomial is \(5\) because there are \(4\) extreme values. Also notice how the graph approaches positive infinity on the left and negative infinity on the right, Since the graph is going in opposite directions at the extremes, it must have an odd degree. The sign of the leading coefficient is negative because as \(x\) approaches positive infinity, \(y\) approaches negative infinity. |
Example 2: Determine the least degree of the polynomial and the sign of the leading coefficient.
Example 3: Without graphing, determine the end behavior of the polynomial \(f\left(x\right)=3x^6-2x^4+5x-7\).
Since the degree of the polynomial is \(6\) (even) and the leading coefficient is positive, the end behavior is
As \(x\ \longrightarrow\ -\infty,\ f\left(x\right)\ \longrightarrow\ +\infty\) and
As \(x\ \longrightarrow\ +\infty,\ f\left(x\right)\ \longrightarrow\ +\infty\)
In words (not symbols) this says as \(x\) approaches negative infinity, \(f\left(x\right)\) approaches positive infinity and as \(x\) approaches positive infinity, \(f\left(x\right)\) approaches positive infinity.
Try to learn how to use the symbols!
Example 4: Without graphing, determine the end behavior of the polynomial \(f\left(x\right)=\frac{1}{5}x^3-2x^2+3x-5\).
Now that we know how the degree and the sign of the leading coefficient affect the graph of a polynomial, we can talk about what the polynomials look like at the x-intercepts. From the graphs you explored at the beginning of the guided learning, you should have noticed that sometimes the graph just touches the x-axis at the x-intercept and other times the graph crosses over the x-axis at the x-intercept.
- If a graph touches the x-axis at the x-intercept, the factor has an even powered exponent. We say it would have an even multiplicity.
- If a graph crosses the x-axis at the x-intercept, the factor has an odd powered exponent. We say it would have an odd multiplicity.
The multiplicity is the number of times the solutions (x-intercept) occurs.
Example 5: Find the roots (solutions) of the polynomial \(f\left(x\right)=2x\left(x-3\right)^3\left(x-1\right)^2\left(x+1\right)\) and indicate the multiplicity of each root.
Find the roots (set each factor equal to zero and solve). The multiplicity is the power of each factor.
\(x = 0\), multiplicity \(1\)
\(x = 3\), multiplicity \(3\)
\(x = 1\), multiplicity \(2\)
\(x = -1\), multiplicity \(1\)
Example 6: The polynomials graphed have the same x-intercepts and they both have an odd degree with a negative leading coefficient. So why do these graphs look different?
The graph on the left is Degree \(3\) (two extreme values). \(x = -1\) has a multiplicity of one because the graph crosses the x-axis at that point, and \(x = 4\) has a multiplicity of two because the graph touches the x-axis at that point. The graph on the right is Degree \(5\). Looking at how the graph behaves at \(x = -1\), you can see that it makes a little dip or flattens out--this is how you can tell it’s multiplicity is odd and greater than \(1\). We call this a point of inflection, which is defined as a point on a curve at which a change in the direction of curvature occurs.
In general, if a graph flattens out at the x-intercept, the multiplicity increases.
Example 7: Sketch the graph of the polynomial with least degree given the following information:
x-intercepts: \(-5\), multiplicity \(2\); \(-1\), multiplicity \(3\); and \(2\), multiplicity \(1\), leading coefficient is positive
Finally, let’s look at all the pieces necessary to analyze a polynomial function. The information that should be included is:
- The degree
- The end behavior
- The x-intercepts and their multiplicity
- The y-intercept
- The maxima and minima
- The intervals where the function is increasing and decreasing
- The domain and range
A couple more definitions (I know, there are so many in this target!):
Domain: The set of \(x\) values (inputs) for which the function is defined (in other words, the function provides a \(y\) value (output) for each \(x\) value (input)). The domain of every polynomial functions is the set of all real numbers.
Range: The set of \(y\) values (outputs) for each \(x\) value in the domain. The range of a polynomial function depends on the degree and the end behavior.
- Odd degree polynomials will have a range of all real numbers.
- Even degree polynomials will have a range that is dependent on the direction the graph opens and the largest maximum or minimum.
Maximum: The highest \(y\) value on a function within an interval of \(x\) values.
- Relative or local maximum--a maximum value but not the largest maximum value
- Absolute or global maximum--a maximum value that is the highest maximum value
Minimum: The lowest \(y\) value on a function, within an interval of \(x\) values.
- Relative or local minimum--a minimum value but not the largest minimum value
- Absolute or global minimum--a minimum value that is the lowest minimum value
Increasing Function: When reading a graph left to right, the interval of \(x\) values (domain) between which the \(y\) values are increasing.
Decreasing Function: When reading a graph left to right, the interval of \(x\) values (domain) between which the \(y\) values are decreasing.
In the next example, we will completely analyze a polynomial. Since we haven’t done any higher order factoring yet, it will be necessary to use the graphing calculator to find the x-intercepts. In addition, we will need to calculate the maximum and minimum values with the calculator. The screen shots for finding the zeros of the function, as well as the maximum and minimum values are shown below.
To find zeros:
To find maximum and minimum:
Example 8: Analyze the polynomial function: \(f\left(x\right)=-3x^4+6x^3+9x^2-24x+12\).