Often we are asked to write a cubic or quartic function from a given graph or from points that will be on the graph. We always start by writing these in intercept (factored) form. Then, depending on how the problem is worded, we will expand to standard form or leave the function written in factored form.
The steps to follow when writing the function are similar regardless of whether the function is cubic or quartic.
1. Find each x-intercept. Include the multiplicity for each x-intercept.
2. Find the minimum degree of a polynomial function, which is the number of turns the graph makes plus one. Keep the degree in mind as you write the factors (especially if the problem does not clearly state the degree).
3. Write the equation in factored form, including their powers of the factors. The degree can help with this, because the sum of the powers of the factors will equal the degree.
4. Solve for the leading coefficient, \(a\), by substituting the values of \(\left(x,y\right)\) into the function.
5. Write the function. Leave in factored form or standard from as indicated by the problem.
Example 1: Write a function in standard form for the cubic polynomial shown.
The steps to follow when writing the function are similar regardless of whether the function is cubic or quartic.
1. Find each x-intercept. Include the multiplicity for each x-intercept.
- If an x-intercept just touches the x-axis (bounces), then the multiplicity is even.
- If an x-intercept crosses the x-axis (passes through), then the multiplicity is odd.
2. Find the minimum degree of a polynomial function, which is the number of turns the graph makes plus one. Keep the degree in mind as you write the factors (especially if the problem does not clearly state the degree).
3. Write the equation in factored form, including their powers of the factors. The degree can help with this, because the sum of the powers of the factors will equal the degree.
- The flatter the graph is at the x-intercept, the higher the power of the factor.
- If the x-intercept just touches the x-axis, the factor is raised to an even exponent.
- If the x-intercept crosses the x-axis, the factor is raised to an odd exponent.
4. Solve for the leading coefficient, \(a\), by substituting the values of \(\left(x,y\right)\) into the function.
5. Write the function. Leave in factored form or standard from as indicated by the problem.
Example 1: Write a function in standard form for the cubic polynomial shown.
\(\begin{align}&4=a\left(2+4\right)\left(2-1\right)^2\ & \ &\text{ 1) Substitute x and y into the above equation and}\\ &\ & \ &\text{solve for a.}\\ &4=a\left(6\right)\left(1\right)^2\\ &4=6a\\ &\frac{4}{6}=a\\ &\frac{2}{3}=a\\ &f\left(x\right)=\frac{2}{3}\left(x+4\right)\left(x-1\right)^2\ \ & \ &\text{2) Write the function in factored form.}\\ &f\left(x\right)=\frac{2}{3}\left(x+4\right)\left(x^2-2x+1\right)\ \ & \ &\text{3) multiply to find the polynomial in standard form.}\\ &f\left(x\right)=\frac{2}{3}\left(x^3+2x^2-7x+4\right)\\ &f\left(x\right)=\frac{2}{3}x^3+\frac{4}{3}x^2-\frac{14}{3}x+\frac{8}{3}\end{align}\) |
Example 2: Write the quartic function with x-intercepts \(x = -2\), multiplicity one; \(x = 2\), multiplicity two; and \(x = 5\), multiplicity one that passes through the point \(\left(0,120\right)\). Leave in factored form.
Example 3: Write the function in standard form for the graph shown:
Writing polynomial functions given irrational conjugate and complex conjugate roots
The expressions \(a+\sqrt{b}\) and \(a-\sqrt{b}\) are called irrational conjugates, and the expressions \(a+bi\) and \(a - bi\) are called complex conjugates, as described in the Quadratics Unit. If you multiply the irrational conjugate expressions or the complex conjugate expressions, you will always get a rational number. The conjugate is formed by changing the sign between the rational number and the radical.
In this section we will write equations given irrational solutions or complex solutions. Before we do so, however, we need to have two theorems.
Irrational Conjugates Theorem: If \(a+\sqrt{b}\) is an irrational root of a polynomial, then its irrational conjugate \(a-\sqrt{b}\) is also a root.
Complex Conjugates Theorem: If \(a+bi\) is a complex root of a polynomial, then its complex conjugate \(a-bi\) is also a root.
Complex Conjugates Theorem: If \(a+bi\) is a complex root of a polynomial, then its complex conjugate \(a-bi\) is also a root.
In other words, irrational roots and complex roots of polynomials functions always come in pairs.
Let’s look at some examples:
Example 4: Write the polynomial function (leading coefficient \(1\)) with roots \(2\) and \(\sqrt{3}\).
Let’s look at some examples:
Example 4: Write the polynomial function (leading coefficient \(1\)) with roots \(2\) and \(\sqrt{3}\).
\(\begin{align}&x=2, x=\sqrt{3}\ \text{and}\ x=-\sqrt{3}\ & \ &\text{ 1) Write the roots (note that you will have to find the other irrational root on your}\\&\ & \ &\text{own--it will not be given to you)}.\\&x-2=0, x-\sqrt{3}=0\ \text{ and}\ x+\sqrt{3}=0\ & \ &\text{2) Get each equation set equal to zero by adding or subtracting the constant from.}\\&\ & \ &\text{both sides.}\\&\left(x-2\right)\left(x-\sqrt{3}\right)\left(x+\sqrt{3}\right)=0\ & \ &\text{3) Multiply the factors together.}\\&\left(x-2\right)\left(x^2+\sqrt{3}x-\sqrt{3}x-3\right)=0\ & \ &\text{4) Multiply the irrational conjugates first.}\\&\left(x-2\right)\left(x^2-3\right)=0\ & \ & \text{5) Multiply the binomials.}\\&f\left(x\right)=x^3-2x^2-3x+6\ & \ &\text{6) Write as a function.}\end{align}\)
Example 5: Write the polynomial function (leading coefficient \(1\)) with roots \(-3\) and \(-\sqrt{5}\).
Example 6: Write the polynomial function (leading coefficient \(1\)) with roots \(7\) and \(-5i\).
\(\begin{align}&x=7, x=-5i\ \text{and}\ x=5i\ & \ &\text{1) Write the roots (note that you will have to find the other complex root}\\& \ & \ &\text{on your own).}\\&x-7=0, x+5i=0\ \text{and}\ x-5i=0\ & \ &\text{2) Get each equation set equal to zero by adding or subtracting the.}\\& \ & \ &\text{constant from both sides.}\\&\left(x-7\right)\left(x+5i\right)\left(x-5i\right)=0\ & \ &\text{3) Multiply the factors together.}\\&\left(x-7\right)\left(x^2-5ix+5ix-25i^2\right)=0\ & \ &\text{4) Multiply the complex conjugates first.}\\&\left(x-7\right)\left(x^2-25\cdot\left(-1\right)\right)=0\ & &\text{5) Remember}\ i^2=-1.\\&\left(x-2\right)\left(x^2+25\right)=0\ & \ &\text{6) Multiply the binomials.}\\&f\left(x\right)=x^3-2x^2+25x-50\ & \ &\text{7) Write as a function.}\end{align}\)
Example 7: Write the polynomial function (leading coefficient \(1\)) with roots \(-3\) and \(4+3i\).
Polynomial Functions of Best Fit (Regression) with the calculator
Cubic and Quartic Regression can also be found using the graphing calculator. You may recall from the Equations and Inequalities Unit Target C or the Quadratics Unit Target F how to enter data into the calculator, but the steps and screenshots are reviewed below.
- The calculator defaults to List 1 and List 2, but you may use any list you want as long as you identify those lists in Step 4 above.
- If you do not have List 1 or List 2, Select 5: SetUpEditor from Step 1 then ENTER.
- Clear all lists before entering new data by highlighting the list name in Step 2 then pressing CLEAR and ENTER.
- The \(R^2\) value is the coefficient of determination. If you do not have this shown, you can turn the diagnostics on by pressing 2nd 0 (catalog menu) and scroll down to DiagnosticOn and press enter. The coefficient of determination can help us determine if the regression equation is a good model (similar to correlation coefficient).
Example 8: An open box is created by cutting \(x\) by \(x\) inch units from each corner of a sheet of cardboard. The dimensions cut from each corner, \(x\), and the corresponding volume, \(y\), are shown. Find the cubic regression model for the data. Find the volume of the box when \(x = 3.5\) inches. Find the maximum volume of the box.