Algebra 2 Graphing With the Quadratic Formula
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The Big Graphing Shortcut
Graphing With the Quadratic Formula

 

 There are some graphing tricks that use the quadratic formula.

 To use them, we first have to set up the equation to look like this:
 

 Y = AX2 + BX + C

 
 Remember that A, B, and C are just placeholders 
 that stand for any old number we might have in the equation.
 
 Now look at the mess of an equation we got at the end of the last page:
 

 

 
 The place where the parabola turns around (called the vertex) is at:
 

 

 
 That means if we have the equation:
 

 Y = 3X2 - 6X + 4

 

 A = 3   B = -6   C = 4

 (Don't forget the signs go with the numbers!)

 
 The place where the parabola turns around is where ...

 

 
 So the graph turns around where X = 1.
 
 Where is Y when X = 1?
 
 The messy equation from the last page says that at that point ...
 

 

 
 It's true. And you can find it that way if you want.
 But you can also just plug X = 1 back into the original problem
 and see what you get for Y.
 

Y = 3X2 - 6X + 4

 

Y = 3(1)2 - 6(1) + 4

 

Y = 3 - 6 + 4

 

Y = 1
 
 So the graph vertex is at X = 1, Y = 1.
 
 Does the graph go up or down?
 Is the graph wide or skinny?
 
 Look at the messy equation from the last page.
 The number that answers those questions is what we are calling "A."
 It's the number next to the X 2.
 In this equation, that number is 3.
 
 That means:
 3 is positive, so the graph goes up.
 3 is bigger than 1 so the graph is skinny.
 And from the stuff we did before, 
 we know the vertex is at X=1, Y= 1.
 
 That's enough info to draw the graph ...

 

 
 Let's try another one ...
 
 Example:
 
 What does the graph of Y = -X 2 + 4X -3 look like?
 
 The first thing to do is figure out what A, B, and C are for this equation.
 B = 4 and C = -3. Those are no problem, but what about A?
 
 Here's the deal, if there is no number next to the X 2 or X
 the number is 1.
 It's really there, we just don't need to write a "times one."
 If there is a minus sign next to the X, the number is -1.
 So:

 A = -1   B = 4   C = -3

 
 And the X value at the vertex:

 

 
 To find Y when X = 2, 
 just substitute 2 into the original equation for X.
 

 

Y = -(2)2 + 4(2) -3 

 

Y = -4 + 8 -3 

 

Y = 1
 
 So  the vertex is at X = 2, Y = 1.
 
 OK, does the parabola open up or down?
 A = -1, so it opens down.
 -1 is still a 1, so the graph is not too thin or wide.
 
 So draw it ...

 

 

   copyright 2005 Bruce Kirkpatrick

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