Graphing Quadratic Functions

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Graphing Quadratic Functions
MA.912.A.7.1 Graph quadratic equations. MA.912.A.
7.6 Identify the axis of symmetry, vertex,
domain, range, and intercept(s) for a given
parabola
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Quadratic Function

• y ax2 bx c

Quadratic Term
Linear Term
Constant Term
What is the linear term of y 4x2 3? 0x
What is the linear term of y x2- 5x ? -5x
What is the constant term of y x2 5x? 0
Can the quadratic term be zero? No!
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Solving a Quadratic
The x-intercepts (when y 0) of a quadratic
function are the solutions to the related
quadratic equation.

• The number of real solutions is at most two.

One solution X 3
Two solutions X -2 or X 2
No solutions
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Quadratic Functions
parabola
The graph of a quadratic function is a
A parabola can open up or down.
If the parabola opens up, the lowest point is
called the vertex (minimum).
If the parabola opens down, the vertex is the
highest point (maximum).
NOTE if the parabola opens left or right it is
not a function!
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Standard Form
The standard form of a quadratic function is
y ax2 bx c
The parabola will open up when the a value is
positive.
The parabola will open down when the a value is
negative.
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Axis of Symmetry
Parabolas are symmetric.
If we drew a line down the middle of the
parabola, we could fold the parabola in half.
We call this line the Axis of symmetry.
If we graph one side of the parabola, we could
REFLECT it over the Axis of symmetry to graph the
other side.
The Axis of symmetry ALWAYS passes through the
vertex.
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Finding the Axis of Symmetry
When a quadratic function is in standard form
y ax2 bx c,
the equation of the Axis of symmetry is
This is best read as the opposite of b
divided by the quantity of 2 times a.
Find the Axis of symmetry for y 3x2 18x 7
The Axis of symmetry is x 3.
a 3 b -18
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Finding the Vertex
The Axis of symmetry always goes through the
_______. Thus, the Axis of symmetry gives us the
____________ of the vertex.
Vertex
X-coordinate
Find the vertex of y -2x2 8x - 3
STEP 1 Find the Axis of symmetry
The x-coordinate of the vertex is 2
a -2 b 8
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Finding the Vertex
Find the vertex of y -2x2 8x - 3
STEP 1 Find the Axis of symmetry
STEP 2 Substitute the x value into the
original equation to find the y coordinate of
the vertex.
The vertex is (2 , 5)
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Graphing a Quadratic Function
There are 3 steps to graphing a parabola in
standard form.
STEP 1 Find the Axis of symmetry using
STEP 2 Find the vertex
STEP 3 Find two other points and reflect them
across the Axis of symmetry. Then connect the
five points with a smooth curve.
MAKE A TABLE using x values close to the Axis
of symmetry.
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Graphing a Quadratic Function
STEP 1 Find the Axis of symmetry
STEP 2 Find the vertex
Substitute in x 1 to find the y value of the
vertex.
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Graphing a Quadratic Function
STEP 3 Find two other points and reflect them
across the Axis of symmetry. Then connect the
five points with a smooth curve.
1
5
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Y-intercept of a Quadratic Function
The y-intercept can be found when x 0.
Y-intercept is at (0, c)
(0, -1)
The constant term is always the y- intercept
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Identifying Solutions
Find the solutions of 2x - x2 0
The solutions of this quadratic equation can be
found by looking at the graph of f(x) 2x x2
The x-intercepts are the solutions to 2x - x2
0 Use factoring or quadratic formula to find x
intercepts
X 0 or X 2
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X and Y-intercepts

• Find the vertex, the y intercept, the symmetric
  point to the y intercept, and the x intercepts
  for the following equation

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Intercepts

• Find the coordinates of the vertex ,
  y-intercepts, symmetric point and x intercepts.