Graphing Quadratic Functions

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Graphing Quadratic Functions
Graph quadratic equations. 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

a - constant
b - constant
c - constant
Note a is the leading coefficient
Note letters a, b, and c are constant terms
Note letters x and y are variables
Can the constant a be zero? No!
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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 (AOS).
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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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).
The vertex is (h, k) where h AOS and k max/min
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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 (AOS).
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 of a Quadratic function can Be
found when x 0.
The constant term is always the y- intercept
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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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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(or Zeros) of
f(x) 2x x2 are the solutions to 2x - x2 0
X 0 or X 2