Graphs from Cubic and Quartic Equations
We have seen quadratic equations, which create parabolas. Recall that quadratic equations are of the form y=ax^2+bx+c. Let's now look at equations with a higher degree or exponent on x. These equations will have a higher "power" of x, specifically we will look at equations that look like y=ax^3+bx^2+cx+d, called cubic equations, and y=ax^4+bx^3+cx^2+dx+e, called quartic equations.
Cubic Equations
Quartic Equations
Quartic equations are of the form y=ax^4+bx^3+cx^2+dx+e.
Visit the following website, which shows an equation that represnts the shape of the Gateway Arch in St. Louis, Missouri.
[1] "The Gateway Arch as a quartic equation"
Answer the following questions related to the website:
1. What equation is used to represent the shape of the Gateway Arch?
2. What qualities of the Gateway Arch is this quartic equation good for representing?
With a partner, analyze what happens when you move the red dots in the manipulative. Do not limit yourself to the shape of the Gateway Arch. Notice that as you move the red dots, the equation in the top right corner will change but it will always have a term with x^4.
Answer the following questions:
1. Draw some different sketches of the different shapes quartic graphs can take.
2. By moving the red dots, try to create a quartic equation with different numbers of roots. Is it possible to get a quartic with one root, two roots, three roots, or four roots? For each example you can find for each of the different numbers of roots, give a sketch of the graph and explain how the graph has the given number of roots. Hint: Recall that a parabola can have a maximum or minimum point - the quartic can actually have more of these points which are maximums or minimums relative to the points around them. What would happen if one of these maximum points was directly on the x-axis?
Reflection
What have you learned about cubic and quartic equations in real world contexts? What is the relation of cubic and quartic equations?