Download § 8.1 Square Root Property and Completing the Square

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§ 8.4 Quadratic Functions and their
Graphs
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Graphs of Quadratic Functions
The graph of a quadratic function has the shape
of a parabola. But, what exactly is parabola?
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Graphs of Quadratic Functions
The graph of a quadratic function has the shape
of a parabola. But, what exactly is parabola?
The curve formed by the set of points in a plane
that are equally distant from both a given
line(called the directrix) and a given point
(called the focus) that is not on the line.
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Graphing Quadratic Functions
𝑓 𝑥 = 𝑥2
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Graphing Quadratic Functions
𝑓 𝑥 = 𝑥2
We need to choose values for 𝑥 to give us 𝑦
values that will help us plot points on a
coordinate graph. In general, choosing values
around 0 helps. So numbers such as -3,-2,1,0,1,2,3.
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Quadratics of the Form 𝑎𝑥2.
1.) If the coefficient of the term with 𝑥 2 is a
positive number, then the parabola will open
upwards.
2.) If the coefficient of the term with 𝑥 2 is a
negative number, then the parabola will open
downwards.
This will be true always no matter what the
coefficients are for the other less degree terms.
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Quadratics of the Form 𝑎𝑥2.
1.) If the coefficient of the term with 𝑥 2 is a
positive number, then the parabola will open
upwards.
2.) If the coefficient of the term with 𝑥 2 is a
negative number, then the parabola will open
downwards.
This will be true always no matter what the
coefficients are for the other less degree terms.
E.g.: Looking at 𝑓 𝑥 = −2𝑥 2 .
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Graphing Quadratic Functions
𝑓 𝑥 = 𝑥2
We need to choose values for 𝑥 to give us 𝑦 values
that will help us plot points on a coordinate graph. In
general, choosing values around 0 helps. So
numbers such as -3,-2,-1,0,1,2,3.
Now looking at 𝑔 𝑥 = 𝑥 2 − 2.
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Graphing Quadratic Functions
𝑓 𝑥 = 𝑥2
We need to choose values for 𝑥 to give us 𝑦 values
that will help us plot points on a coordinate graph. In
general, choosing values around 0 helps. So
numbers such as -3,-2,-1,0,1,2,3.
Now looking at 𝑔 𝑥 = 𝑥 2 − 2.
Now looking at ℎ 𝑥 = (𝑥 − 2)2 .
What is the difference?
Combinations of Transformations
Let’s look at,
𝑠 𝑥 = (𝑥 + 3)2 + 4
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Combinations of Transformations
Let’s look at,
𝑠 𝑥 = (𝑥 + 3)2 + 4
The graph of 𝑠 𝑥 is just the graph of 𝑥 2 , but
shifted to the left 3 units and raised 4 units up.
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Combinations of Transformations
Let’s look at,
𝑠 𝑥 = (𝑥 + 3)2 + 4
The graph of 𝑠 𝑥 is just the graph of 𝑥 2 , but
shifted to the left 3 units and raised 4 units up.
I like to have the notion of “outside” and “inside” of
the parenthesis. The 3 is “inside” and the 4 is
“outside”.
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Combinations of Transformations
Let’s look at,
𝑠 𝑥 = (𝑥 + 3)2 + 4
The number that you add/subtract in the “inside”
will shift the graph of 𝑥 2 to the left/right that many
units while the number that you add/subtract in the
“outside” will shift the graph of 𝑥 2 up/down that
many units.
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Combinations of Transformations
Let’s look at,
𝑠 𝑥 = (𝑥 + 3)2 + 4
The quadratic equation here is in what is called
Standard Form. Basically, it is in the form,
𝑓 𝑥 = 𝑎(𝑥 − ℎ)2 +𝑘
Where our 𝑎 = 1, ℎ = −3, 𝑘 = 4.
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Combinations of Transformations
Let’s look at,
𝑠 𝑥 = (𝑥 + 3)2 + 4
The quadratic equation here is in what is called
Standard Form. Basically, it is in the form,
𝑓 𝑥 = 𝑎(𝑥 − ℎ)2 +𝑘
Where our 𝑎 = 1, ℎ = −3, 𝑘 = 4. The vertex of a
parabola is the point with coordinates ℎ, 𝑘 . It is
basically the lowest/highest point on the parabola.
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Axis of Symmetry
The Axis of symmetry has the equation,
𝑥=ℎ
Where ℎ comes from the equation
𝑓 𝑥 = 𝑎(𝑥 − ℎ)2 +𝑘.
It is the vertical line that basically “cuts the
parabola is half”.
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Vertex Formula
To get the 𝑥-coordinate of the vertex point from
the quadratic in the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, it’s just
−𝑏
𝑥=
2𝑎
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Vertex Formula
To get the 𝑥-coordinate of the vertex point from
the quadratic in the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, it’s just
−𝑏
𝑥=
2𝑎
To get your 𝑦-coordinate, just plug in your result
for 𝑥 back into the quadratic equation.
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Vertex Formula
To get the 𝑥-coordinate of the vertex point from
the quadratic in the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐, it’s just
−𝑏
𝑥=
2𝑎
To get your 𝑦-coordinate, just plug in your result
for 𝑥 back into the quadratic equation.
Let’s graph,
𝑓 𝑥 = 𝑥 2 + 6𝑥 + 5.
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Number of Real Solutions
 If the graph of a parabola intersects the x-
axis twice, then the quadratic equation has 2
real solutions.
 If the graph of a parabola intersects the xaxis at a single point, then the quadratic
equation has 1 real solution.
 If the graph of a parabola does not intersect
the x-axis at all, then the quadratic equation
has no real solutions.
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Graphing functions of the form 𝑎𝑥2 + 𝑏𝑥 + 𝑐
by Completing the Square
We will need to use completing the square for a
quadratic function of the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 to get it in
the form 𝑎(𝑥 − ℎ)2 +𝑘.
E.g.:
3𝑥 2 − 12𝑥 + 8
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Sketch the graph
𝑓 𝑥 = −2𝑥 2 + 12𝑥 − 10
1.) Find the coordinates of the vertex.
2.) What is the axis of symmetry?
3.) Is it going to open upward or downward?
4.) Choose values for x around the h value of
the vertex and make a “T table” to get points.
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Sketch the graph
𝑓 𝑥 = −2𝑥 2 + 12𝑥 − 10
1.) Find the coordinates of the vertex.
2.) What is the axis of symmetry?
3.) Is it going to open upward or downward?
4.) Choose values for x around the h value of
the vertex and make a “T table” to get points.
Also, recall, that to find the x-intercepts, set the
quadratic function equal to 0 and to find the yintercept, set the x’s equal to 0.