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Graphs of Quadratic Functions
The graph of a quadratic function is called a parabola. There are two key pieces of information needed to graph a parabola. You need to know the vertex, which is the high or low point of the graph, and the roots, which is where the graph crosses or touches the x–axis. To learn more about finding the vertex of a parabola, click here for the vertex lesson. To learn more about finding the roots, click here to go to a lesson on solving a quadratic. In addition to knowing the vertex and the roots, it will also be helpful to plot specific points to get a more accurate graph. To learn more about graphing parabolas on the calculator, click here to go to a lesson on graphing functions.

Examples
Example
#1: Graph
  • Step 1: Find the vertex.




So the vertex is at (-3, -1)
  • Step 2: Find the roots.
Start with

Next, factor the function , set its factors equal to zero, and solve for x.



or

Telling us that x = -4 and x = -2
  • Step 3: Determine if the parabola opens up or down.
Since the coefficient of is positive 1, the parabola opens up.
  • Step 4: Find other points to fill in the graph of
x y
-2 0
-1 3
0 8
1 15
What is your answer?
 
Example
#2: Graph
  • Step 1: Find the vertex.




So the vertex is (1, 1)
  • Step 2: Find the roots.
Since does not factor, we must use the quadratic formula.









The approximate values of roots are x = 0.42 and x = 1.58.
  • Step 3: Determine if the parabola opens up or down.
Since the coefficient of is negative 3, the parabola opens down.
  • Step 4: Find other points to fill in the graph of .
x y
1 -11
0 -2
1 1
2 -2
3 -11
What is your answer?
 
Example
#3: Graph
  • Step 1: Find the vertex.




So the vertex is
  • Step 2: Find the roots.
Since does not factor, we must use the quadratic formula.









These are imaginary roots and the parabola does not touch or cross the x–axis.
  • Step 3: Determine if the parabola opens up or down.
Since the coefficient of is positive 2, the parabola opens up.
  • Step 4: Find other points to fill in the graph of .
x y
-1 2
0 3
1 3
2 7
3 15
What is your answer?
 



S Taylor

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