This activity is intended to supplement Algebra I, Chapter 10, Lesson 7.
Problem 1 – Match the Graph, Part 1
The vertex form for the equation of a parabola is y=a(x−h)2+k. If needed, graph y=a(x)2 with various values of a and explore.
- In vertex form or in standard form, what happens when 0<a<1?
- If a>1, the graph will be narrow and open up. If a<−1, the graph will be what?
Enter the lists shown at the right. Create a scatter plot of L1 and L2. Then, enter the vertex form of the parabola in Y1 with an initial guess for each value for a,h, and k. See how the equation fits and then adjust the values to make the graph fit the data.
- What is the vertex of the parabola?
- What was your value of a for the parabola?
- What is the equation of the parabola you fit to the data?
Problem 2 – Match the Graph, Part 2
Repeat the process from Problem 1 to find the equation of a parabola that matches the data in L1 and L2.
- To make the parabola open down, what must be true about the value of a?
- To make the parabola wide, what must be true about the value of a?
- What is the equation of your parabola that fits the data?
Problem 3 – Match the Double Arches
Change L1 and L2 to match the screenshot shown a the right. Now graph, Y1=(−1.5(X+2)2+5.5)(−4≤X and X≤0)
Next, match the second half of double arches.
- What do you notice about the two parabolas that formed the double arches?
- The vertex of the left arch is (−2,5.5). What is the vertex of the right arch?
- What is the equation of your parabola that matches the data?
Problem 4 – The Main Cables of a Suspension Bridge
Here is a picture of a suspension bridge. Several loops of cable are represented. See the graph below to match an equation to a particular part of the graph.
The point where pieces A and B meet is (0,3.2).
- What is the equation of the piece of the graph labeled A?
- What is the equation of the piece of the graph labeled B?
Extension – The Gateway Arch in St. Louis
The Gateway Arch in St. Louis, the “Gateway” to America, is a shape that looks like a parabola to the casual observer.
Use what you know about the vertex form to write an equation to match its shape and dimensions. Enter L1 and L2 shown and create a scatter plot with an appropriate window.
Using the same data, match the graph in standard form (y=ax2+bx+c) by changing the Y= equation. Important things to remember are; what does the value of a do to the graph, and what would your y−intercept be (c in the equation)?
- What is your equation in standard form?
- How are the two equations similar?
- How are the two equations different?
- Expand the vertex form and convert it to standard form to make a final comparison.