4.1: Graphs of Quadratic Equations in Standard Form Assignment:

4.1: Graphs of Quadratic Equations in
Standard Form
1.
2.
3.
4.
Objectives:
To find the intercepts, vertex,
axis of symmetry, and the
minimum/maximum value of a
parabola
To determine the effect of
changing the parameters of a
quadratic equation
To find the
minimum/maximum value of a
parabola
To graph a quadratic
equation in standard form
Assignment:
• P. 240-243: 1-3, 8, 10, 12,
19, 20, 21-39 M3, 40, 4446, 53, 58, 61-63
• Challenge Problems: 1-6
Vocabulary
Parabola
Parent Function
Vertex
Axis of Symmetry
𝑥-intercepts
𝑦-intercepts
Maximum
Minimum
Perpendicular Bisector
Objective 1
You will know how
to find the
intercepts, vertex,
and the axis of
symmetry of a
parabola
Warm-Up
1. On a piece of
patty paper, use a
ruler to draw AB.
Warm-Up
2. Now fold the piece
of paper so that
point A lies
coincides with (lies
directly on top of)
point B.
Warm-Up
3. Unfold the paper
and label point M
where the crease
intersects the
segment. This
crease is called the
perpendicular
bisector.
Warm-Up
The graph of a quadratic
function is a U-shape
called a parabola.
The x-intercepts are
the points where the
parabola intersects
the x-axis
Warm-Up
The graph of a quadratic
function is a U-shape
called a parabola.
Fold your parabola
to create the
perpendicular
bisector of the
segment connecting
the two x-intercepts
Warm-Up
The graph of a quadratic
function is a U-shape
called a parabola.
This fold line is
called the axis of
symmetry
Warm-Up
The graph of a quadratic
function is a U-shape
called a parabola.
Notice that the axis
of symmetry passes
through the vertex
of the parabola
Standard Form
A quadratic function in standard form is
written y = ax2 + bx + c, a ≠ 0.
a, b, and c are considered
parameters
The graph is a parabola
Parent Function: 𝑦 = 𝑥 2
Parts of a Parabola
A quadratic function in standard form is
written y = ax2 + bx + c, a ≠ 0.
Vertex: The turning point of
the parabola; marks the
highest or lowest point
Vertex
Parts of a Parabola
Axis of Symmetry: Line
through the vertex that
divides the parabola into 2
mirror images
Axis of Symmetry
A quadratic function in standard form is
written y = ax2 + bx + c, a ≠ 0.
Exercise 1
1. What is the domain for any quadratic
function?
2. Explain how the vertex of a parabola
affects the range of the function.
Exercise 2
If the vertex of a parabolic function is (h, k),
what is the equation of the axis of
symmetry?
Exercise 3
Let’s say the vertex of
a parabola is (2, -2).
If the y-intercept of
the parabola is
(0, 6), what other
point must lie on the
parabola
Axis of Symmetry
For a given parabola
with 2 x-intercepts,
consider a segment
connecting these two
zeros.
What is the relationship
between the axis of
symmetry and this
segment?
Axis of Symmetry
The axis of symmetry is
the perpendicular
bisector of the
segment connecting
the zeros.
Exercise 4
Find the 𝑥-intercepts of the
quadratic equation, then use
them to help you find the
coordinates of the vertex
and the axis of symmetry.
𝑦 = 𝑥 2 − 8𝑥 − 12
Exercise 5
Since the axis of symmetry bisects the
segment connecting the zeros of a
parabolic function, use the quadratic
formula to derive the equation of the axis of
symmetry and the coordinates of the vertex
of y = ax2 + bx + c.
Exercise 5
Average of the 𝑥-intercepts:
1 −𝑏 − 𝑏 2 − 4𝑎𝑐 −𝑏 + 𝑏 2 − 4𝑎𝑐
∙
+
=
2
2𝑎
2𝑎
−
𝑏2
− 4𝑎𝑐
2𝑎
+
𝑏2
− 4𝑎𝑐
2𝑎
1 −𝑏 − 𝑏 2 − 4𝑎𝑐 − 𝑏 + 𝑏 2 − 4𝑎𝑐
∙
=
2
2𝑎
1 −2𝑏
∙
=
2
2𝑎
−𝑏
2𝑎
𝑥=−
𝑏
2𝑎
Exercise 5
𝑦-coordinate of the vertex:
𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑏
𝑦=𝑎 −
2𝑎
𝑏 2 − 4𝑎𝑐
−
2𝑎
𝑏 2 − 4𝑎𝑐
+
2𝑎
2
𝑏
+𝑏 −
+𝑐
2𝑎
𝑏2
𝑏2
𝑦=𝑎 2−
+𝑐
4𝑎
2𝑎
𝑏2 2𝑏2
𝑦=
−
+𝑐
4𝑎 4𝑎
𝑏2
𝑦=𝑐−
4𝑎
𝑥=−
𝑏
2𝑎
𝑏
𝑏2
− ,𝑐 −
2𝑎
4𝑎
Vertex and Axis of Symmetry
For the graph of y = ax2 + bx + c:
Vertex
Axis of Symmetry
 b
b2 
 , c 
4a 
 2a
x
b
2a
Exercise 6
Find the coordinates of the vertex and the equation
of the axis of symmetry for each of the following
quadratic equations.
1. 𝑦 = 3𝑥 2 − 24𝑥 + 53
2. 𝑦 = −2𝑥 2 − 32𝑥 − 135
Objective 2
You will be able to determine the
effect of changing the parameters
of a quadratic equation
Exercise 7
With out the use of a
graphing calculator,
graph the quadratic
parent function y = x2
by using a table of
values.
x
y
−3
−2
−1
0
1
2
3
Quadratic Parent Function
What we have just graphed
is the quadratic parent
function: y = x2
A quadratic in standard
form is y = ax2 + bx + c
Changing a, b, and c
just change the graph
of the parent function
Parabolic Parameters
Use the GSP
demonstration to
determine the role of
the parameters a, b,
and c on the graph of
y = ax2 + bx + c.
Beard or Mustache
The “a” value
determines
whether your
parabola is a
beard or a
mustache.
𝒂>0
𝒂<0
Parabolic Parameters
For the graph of y = ax2 + bx + c :
Parameter
Behavior on the Graph
a
1. Determines width of parabola
• As |a| increases, the parabola gets narrower
• As |a| decreases, the parabola flattens out/gets
wider
2. Determines upward or downward shape
• If a > 0,  (beard U)
• If a < 0,  (mustache H)
c
1. Determines the y-intercept of the parabola
• As c increases, the parabola moves up
• As c decreases, the parabola moves down
Exercise 8
Compare each graph to the quadratic parent
function y = x2.
1. y = −4x2
2. y = − x2 – 5
3. f(x) = (1/4)x2 + 2
Objective 3
You will be able to find the
minimum or maximum
value of a parabola
Minimum and Maximum
The vertex of a parabola determines the
minimum or maximum value for the
function.
Exercise 9a
Tell whether the function y = −2x2 + 4x + 3
has a minimum or a maximum value. Then
find the minimum or maximum value.
Exercise 9b
Tell whether the function y = 4x2 + 16x – 3
has a minimum or a maximum value. Then
find the minimum or maximum value.
Objective 3
You will know how to graph a
quadratic equation in standard form
Graphing a Parabola
1.
2.
3.
4.
To graph y = ax2 + bx + c:
Use the a-value to determine  (beard U)
or  (mustache H)
Plot the vertex and axis of symmetry
Plot the x-intercepts and the 𝑦-intercept
(plus its reflection across the axis of
symmetry)
If needed, plot at least one more point and
its reflection across the axis of symmetry
Exercise 11
Graph y = −x2 + 6x + 8
Exercise 11
Graph y = −x2 + 6x + 8
Exercise 12
Graph the function. Label the vertex and the
axis of symmetry.
1. y = x2 – 2x – 1
2. y = 2x2 + 6x + 3
3. f(x) = −(1/3)x2 – 5x + 2
4.1: Graphs of Quadratic Equations in
Standard Form
1.
2.
3.
4.
Objectives:
To find the intercepts, vertex,
axis of symmetry, and the
minimum/maximum value of a
parabola
To determine the effect of
changing the parameters of a
quadratic equation
To find the
minimum/maximum value of a
parabola
To graph a quadratic
equation in standard form
Assignment
• P. 240-243: 1-3, 8,
10, 12, 19, 20, 2139 M3, 40, 44-46,
53, 58, 61-63
• Challenge
Problems: 1-6
“Daddy, where’s your doctor mustache?”