Putting It Together: Power and Polynomial Functions
At the start of this module, you were given the challenge of designing a roller coaster given a set of criteria. Now that you know a bit about polynomials, let’s take another look at the coaster in terms of a polynomial as a function of time, [latex]t[/latex].- The starting height of the coaster must be 200 ft, so [latex]f(0)=200[/latex].
- At [latex]t=3[/latex], the roller coaster goes below ground level, (3, 0).
- At [latex]t=5[/latex], the roller coaster returns to ground level, (5, 0).
- At [latex]t=10[/latex], the roller coaster goes below ground level again, (10, 0).
[latex]f\left(t\right)=a\left(t-3\right)\left(t-5\right)\left(t-10\right)[/latex]
[latex]f\left(t\right)=a\left(t3-18t^2+95t-150\right)[/latex]
[latex]f\left(0\right)=-150a-200[/latex]
[latex]a=-4/3[/latex]
[latex]f(t)=-4/3(t3-18t^2+95t-150)[/latex]
Graphing this polynomial will help you analyze the design better. One way to graph a polynomial function is to first find the intercepts. Earlier you found the intercepts [latex](0, 200), (3, 0), (5, 0)[/latex], and [latex](10, 0)[/latex].![The graph begins at the point (0, 200) and then slopes downward. It passes the x-axis through the point (3, 0), goes down to a local minimum, then rises back up through the point (5, 0). It reaches a local maximum and then slopes downward to again pass through the x-axis at (10, 0).](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2017/03/17182247/polynomial-graph-198x300.png)
![The graph begins at the point (0, 200) and then slopes downward. It passes the x-axis through the point (3, 0), goes down to a local minimum, then rises back up through the point (5, 0). It reaches a local maximum and then slopes downward to again pass through the x-axis at (10, 0). Label at (0, 200) says “Begins at 200 ft.” Label says “Goes below ground level” with arrows pointing to (3, 0), and (10, 0). Label says “Comes back above ground level: and points to (5, 0).](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2017/03/17184104/polynomial_roller-coaster-198x300.png)
Licenses & Attributions
CC licensed content, Original
- Putting It Together: Power and Polynomial Functions. Authored by: Lumen Learning. License: CC BY: Attribution.
- Polynomial Graph. Authored by: Christine Caputo for Lumen. License: CC BY: Attribution.
- Polynomial Graph with Roller Coaster. Authored by: Lumen. License: CC BY: Attribution.