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Activities for Intermediate Algebra: Instructors’ Guide

Katherine Yoshiwara

Section 4.3 Curve Fitting

In this Lesson we use a linear system to fit a parabola through three points and use quadratic regression to fit a model to data. The Activities do not include quadratic regression; if you wish to cover that topic you could use Exercises 4.41 and 4.45 in the text as classroom examples.

Activity 4.3.1. Using Three Points.

You are driving at 60 miles per hour when you step on the brakes. Find a quadratic formula, \(~d=at^2+bt+c\text{,}\) for the distance in feet that your car has traveled \(t\) seconds after braking.
\(t\) (sec) \(1\) \(2\) \(3\) \(4\)
\(d\) (feet) \(81\) \(148\) \(210\) \(240\)
  1. Use the first three data points from the table to write three equations.
  2. Simplify the equations, and write a system.
  3. Solve the system.
  4. State the solution to the problem.

Activity 4.3.2. Finding the Equation of a Parabola.

We now have three forms for the equation of a parabola:
  • standard form \(~~~y=ax^2+bx+c~\)
  • factored form \(~~~y=a(x-r_1)(x-r_2)~\)
  • vertex form \(~~~y=a(x-x_v)^2+y_v~\)
  1. Each of these forms is useful for highlighting different properties of the parabola.
    1. What is the \(y\)-intercept of the graph of \(~y=3x^2-5x+8~\text{?}\)
    2. What are the \(x\)-intercepts of the graph of \(~y=2(x-3)(x+5)~\text{?}\)
    3. What is the vertex of the graph of \(~y=4(x+2)^2-5~\text{?}\)
  2. How can we find the equation of a parabola?
    • If we know the \(x\)-intercepts of the parabola, then we use the factored form.
    • If we know the vertex, we use the vertex form.
    • In each case, we must know one other point on the parabola in order to find the value of \(a\text{.}\)
    1. Problem 1
      1. Write an equation for a parabola that has \(x\)-intercepts at \((2,0)\) and \((-3,0)\text{.}\) (There are many possible answers.)
      2. Write an equation for another parabola that has the same \(x\)-intercepts.
      3. Write an equation for the parabola that has \(x\)-intercepts at \((2,0)\) and \((-3,0)\) and \(y\)-intercept at \((0,3)\text{.}\)
    2. Problem 2
      1. Write an equation for a parabola whose vertex is the point \((-2,6)\text{.}\) (Many answers are possible.)
      2. Find the value of \(a\) if the \(y\)-intercept of the parabola is 18.
    3. Problem 3
      Find an equation for each parabola. Use the vertex form or the factored form of the equation, whichever is more appropriate.

Activity 4.3.3. Curve-Fitting with the Vertex Form.

Francine is designing a synchronized fountain display for a hotel in Las Vegas. For each fountain, water emerges in a parabolic arc from a nozzle three feet above the ground. Francine would like the vertex of the arc to be eight feet high, and two feet horizontally from the nozzle.
  1. Choose a coordinate system with its origin at the base of the fountain. Label the coordinates of two points on the path of the water.
  2. Write an equation for the path of the water.
  3. How far from the base of the nozzle will the stream of water hit the ground?

Subsection 4.3.1 Check Your Understanding

  1. What were the variables in your 3x3 system in Activity 1?
  2. If you use the last three points in the table instead of the first three points, will you get the same regression equation? Why or why not?
  3. Name three forms for quadratic equations.
  4. Why do we need to know a second point besides the vertex when using the vertex form to find the equation of a parabola?

Subsection 4.3.2 Wrap-Up

In this Lesson, we worked on the following skills and goals related to quadratic models.
  • Use Gaussian elimination to fit a parabola to three points
  • Choose the best method to find the equation for a parabola
  • Find the equation for a parabola, given its graph

Subsection 4.3.3 Questions for Writing or Discussion

  1. Discuss the three forms for quadratic equations, and when each is useful.
  2. Explain how to fit a parabola through three points.
  3. How do we choose an appropriate type of equation to fit a data set?
  4. If you use the last three points in a collection of data instead of the first three points, will you get the same regression equation? Explain why or why not.

Concept Questions.

  1. Which statement is true?
    1. A parabola is a good model for any non-linear curve.
    2. We will get the same equation by fitting a parabola through any three points of a data set.
    3. We can use Gaussian elimination to fit a parabola through three points.
    4. When we use quadratic regression, the lowest (or highest) point of a data set will be the vertex of the parabola.
  2. How many data points must a quadratic regression equation pass through?
    1. One
    2. Two
    3. Three
    4. None
  3. If you know the \(x\)-intercepts of a parabola, how many more points do you need to find its equation?
    1. One
    2. Two
    3. Three
    4. None
  4. If you know the vertex of a parabola, how many more points do you need to find its equation?
    1. One
    2. Two
    3. Three
    4. None
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