Parabolas: Getting to the Point
In this investigation you will explore what is called the "Vertex Form" of a quadratic function (we will, during this worksheet, by convention refer to these quadratic functions as "parabolas")

  1. Introduction

    Suppose f(x) = 3(x-1)2+2. You could "multiply out" (we call this "using FOIL", but really you would be using the distributive property, which should be getting the credit...) in order to re-write f(x) in the standard, Ax2+Bx+C format (where A, B, and C are constants, and each is positive or negative or zero). But to answer the initial set of questions, you don't need to; leave it as is.

    1. What is f(1) equal to?__________.

    2. Now let's look at another quadratic, this time g(x) = q(x-a)2+b. This is still a quadratic function of x, only we don't know what the values of the constants "q" and "a" and "b" are, but that's ok. What is g(a)? __________

    3. If you had to explain to a student how you know that g(x) is a quadratic function (the student knows that by definition, a quadratic function of x is one of the form Ax2+Bx+C where A, B, and C are constants), what would you say? ________________________________________.

    4. Which of the constant(s) "a", "b" and "q" determine whether the graph of g(x) is a parabola which is U-shaped or one which is an "upside down"? What will determine which shape the graph will have? __________________________________________________.

    5. Fill in the blanks.
      If __________ then q(x-a)2 + b  >  b

      If __________ then q(x-a)2 + b  <  b

    6. The point (a,b) is the highest point on the parabola (the graph of g(x) above) if __________ and it is the lowest point if __________.

    7. Explain why the Vertex Form is important and useful. Hint: suppose you know that the number of lives saved, if you intervene in an endangered ecosystem, is -3x2 + 200x -12, where "x" is a variable that represents how you intervene (we will see an example next time). Is it obvious what x should be? _____ Now suppose that you know that the number of lives saved is -40(x-20)2+2300 What should x be, to save the greatest number of lives? __________. (And, how many lives would be saved? __________).

      Now you know why it's worth the effort to learn how to put quadratics into Vertex Form (the q(x-a)2+b form) Applications will be shown next time.

  2. Looking at Some Examples

    1. On the back, draw a rough qualitative sketch of the graph of the function y = -2(x-1)2+3 (we




      implicitly define a function f(x) = -2(x-1)2+3 this way) 
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    2. In Pairs: Each of you should make a rough sketch of a parabola, so one can see where its vertex (the bottom of the "U" shape or the top of the "upside down U" shape). Now your partner will write a formula for the function whose graph is the parabola you sketched, with at most one "unknown constant" Repeat this several times, and/or start with formula first instead of starting with the graph, until you're comfortable.

    3. Suppose the vertex of a parabola is at the point (1,3). Suppose that you also know that the parabola passes through (3,11). Together come up with a vertex-form formula for the quadratic whose graph is given Discuss this in pairs until you are clear and comfortable with this method

  3. How to find the Vertex Form (Feel free to work through this in groups of 3 or more, until the activity in pairs is mentioned)

    1. Factor out the leading coefficient. Ax2+Bx+C becomes A(x2+Dx+E)

    2. Now how do you re-write (x2+Dx+E) into vertex form (notice that once this is in vertex form, you can multiply by that outside A and the result will be a different parabola, but one which IS in vertex form too). Let W equal (D/2)2. We will see in a minute why W stands for "What We Want"

    3. Re-write (x2+Dx+E) as (x2+Dx+W) + (E-W). Make sure you understand why this is legal

    4. FACT: (x2+Dx+W) is a perfect square. In this step you re-write (that is, factor) it as (x-a)*(x-a) for some "a".

    5. Notice: you now have re-written (x2+Dx+E) as (x2+Dx+W) +(E-W) and you've re-written the latter as (x-a)2 + (E-W).

    6. Your original function was A(x2+Dx+E) which therefore is equal to A[(x-a)2 + (E-W)] Thus this is in Vertex Form "q(x-a)2 + b" if we just let q=_____ and b=_____

    7. Now practice this procedure on 2x2 + 8x + 10.

    8. In pairs, make up problems for one another; your partner's job is to turn your "regular form" quadratic into a Vertex Form quadratic (suggestion: to avoid "ugly/messy numbers", you can privately start with a Vertex Form quadratic with integer coefficients, square it out, and get a "regular form" quadratic, and only then show it to your partner).