Axis of Symmetry
Axis of Symmetry
There is formula for finding the axis of symmetry of a quadratic equation:
y = ax2 + bx + c, 
Example: y = x2 + 4x + 1
Steps:
List a, b: a = ___ b = ___
Answer: a = 1, b = 4
Plug into the formula: = 
Simplify: x = ___
Answer: x = -2
Therefore, the axis of symmetry is x = - 2
The Vertex
Did you notice anything from the previous problems?
When the vertex is (-1, -4) the axis of symmetry is x = -1.
When the vertex is (3, -4) the axis of symmetry is x = 3.
When the verted os (-4, -4) the axis of symmetry is x = -4.
So if we have the axis of symmetry, we have one value of the vertex. All we need to do is evaluate the equation and we will have the other value.
Steps: Find the axis of symmetry
We found this in the previous problem: x = -2
Plug in the x value: y = ( ___ )2 + 4( ___ ) + 1
Answer: -2, -2
Simplify: y = ___ - ___ + 1 = ___
Answer: 4, 8, -3
The vertex is ( ___, ___ )
Answer: -2, -3
The x and y Intercepts
It is easy to find or at least estimate the x and y intercepts when graphing. How does this connect algebraically?
Look at the graph of y = x2 + 5x + 6
Name the y intercept: y = ___
Answer: 6
Name the x intercepts: x = ___ and x = ___
Answer: -3, -2
Notice the y intercept is c in the equation. If we move the -3 and the -2 to the left hand side we would have x + 3 = 0 and x + 2 = 0.
These are factors of:
y = x2 + 5x + 6
(x + 3)(x + 2) = 0
Example: Using the equation y = x2 - 4x - 5, find the c and y intercepts, the axis of symmetry and graph.
In this equation, a = ___, b = ___, c = ___
Answer: 1, -4, -5
This means the y intercepts is y = ___
Answer: -5
The x intercepts are found by factoring:
(x - 5)(x + 1) = y and solving x - 5 = 0 and x + 1 = 0 so the x intercepts are x = ___ and x = ___.
Answer: 5, -1
The axis of symmetry: 
Answer: -4/1
Simplify and the axis of symmetry is x = ___.
Answer: 2
Plug in x to get the rest of the vertex:
y = ( ___ )2 -4( ___ ) - 5 = ___
Answer: 2, 2, -9
Vertex is ( ___, ___ )
Answer: 2, -9
Now let's graph.
The y intercept is y = -5.
The x intercepts are x = 5 and x = -1.
What was the axis of symmetry? x = 2
Vertex is (2, -9).
Now complete the parabola.
