M10 L1

Quadratic Function

Graphing Quadratic Functions| M10_L1

Benchmarks

• MA.912.AR.3.6. Given an expression or equation representing a

quadratic function, determine the vertex and zeros and
interpret them in terms of a real-world context.

• MA.912.AR.3.7 Given a table, equation or written description of

a quadratic function, graph that function, and determine and
interpret its key features.

• MA.912.AR.3.8

Solve and graph mathematical and real-world

problems that are modeled with quadratic functions. Interpret
key features and determine constraints in terms of the context.

• MA.912.F.1.6 Compare key features of linear and nonlinear

functions each represented algebraically, graphically, in tables
or written descriptions.

Goal

•I can graph and analyze quadratic functions.

Essential Questions

•How can you use the equation of the parabola when

the value of a is negative?

•How can you use the equation of a quadratic function

to visualize its graph?

Check for Understanding

Charger Checks, Exit Ticket

​Quadratic Equations by Shm https://youtu.be/BGz3pkoGPag oop

❑Standard Form of a

Quadratic Function is

𝑎𝑥2+𝑏𝑥 + 𝑐

❑A second-degree

polynomial. For
example, 𝑥2+ 2𝑥 − 8, is
a quadratic polynomial,
which has a related
quadratic function
𝑓(𝑥) = 𝑥2+ 2𝑥 − 8

❑The graph of a

quadratic function is
called a parabola

Function

Quadratic

Vocabulary

❑ The axis of symmetry

intersects a parabola
at the vertex.

❑ The vertex is either the

lowest point (minimum)
or the highest point
(maximum) on a
parabola.

❑Parabolas are

symmetric about a
central line called
the axis of symmetry
whose formula is

𝑥 = −𝑏

2𝑎

(1, −4)

𝑥 = 1

(−1, 0)
(3, 0)

(0, −3)

Vertex

Axis of

Symmetry

y-intercept

𝒙 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕
𝒙 − 𝒊𝒏𝒕𝒆𝒓𝒄𝒆𝒑𝒕

1. Identify the axis of symmetry,
the vertex, and the y-intercept
of the graph. Then describe the
end behavior.

❑ The axis of

symmetry is 𝑥 = 0.

❑ The vertex is located

at the maximum
point, (0, 1).

❑ The y-intercept is 1.

1. Identify the axis of symmetry,
the vertex, and the y-intercept
of the graph. Then describe the
end behavior.

❑ As x increases, y

decreases,

❑ As x decreases, y

decreases.

2. Identify the axis of symmetry,
the vertex, and the y-intercept
of the graph. Then describe the
end behavior.

Open Ended

What is the axis of symmetry?

Open Ended

What is the y-value of the Vertex?

2. Identify the axis of symmetry,
the vertex, and the y-intercept
of the graph. Then describe the
end behavior.

❑ The axis of symmetry

is 𝑥 = −3.

❑ The vertex is located

at the maximum point,
(−3, −6).

❑As x increases, y

increases

2. Identify the axis of symmetry,
the vertex, and the y-intercept
of the graph. Then describe the
end behavior.

❑The y-intercept is 3.

❑As x decreases, y

increases.

3. Identify the axis of symmetry, the vertex, and the y-intercept of
the graph. Then describe the end behavior.

Draw

Complete

Poll

Consider the previous question, does the parabola have a maximum or a minimum?

• minimum

• maximum

Graphing Quadratic Functions Using Key Features

Step 1

Find the axis of symmetry, 𝑥 = −

𝑏
2𝑎.

Step 2

Find the vertex and determine whether it
is a maximum or minimum.

Step 3

Find the y-intercept.

Step 4

Use symmetry to find additional points on
the graph, if necessary.

Step 5

Connect the points with a smooth curve.

4. Graph

𝑓 𝑥 = 3𝑥2− 6𝑥 − 1

❑ vertex: (1, −4)

❑ y-intercept: −1

❑ a point is (3,8)

❑ Because the 𝑎 value

in the equation is
positive 3, the graph
will open up.

❑ axis of symmetry: 𝑥 = 1

Draw

Complete #5

Draw

Complete

​Complete #6 in your Cornell Notes

Consider the previous
question, why is the
graph only shown in
Quadrant I?

Step 4

Interpret the key features.

❑ zeros: x = 6.4, the projectile

landed about 6.4 seconds
after it was launched.

❑ y-intercept: The y-intercept

is (0, 40), the projectile is
launched from a height of
40 feet.

❑ vertex: (3, 184), the

projectile reached max
height of 184 feet, 3
seconds after launch.

Step 4

Interpret the key features.

❑ symmetry: The line of symmetry

is x = 3, so the height of the
projectile from 0 to 3 seconds
after launch is the same as the
height from 3 to 6 seconds after
launch.

❑ increasing: The function is

increasing for x < 3, so it is
gaining height up to 3 seconds
after launch.

❑ decreasing: The function is

decreasing for x > 3, so it is
falling 3 seconds after launch

Step 4

Interpret the key features.

❑ positive: The function is

positive for 0 ≤ 𝑥 < 6.4, so
the height of the projectile was
positive from 0 to 6.4 seconds
after launch.

❑ end behavior: As x increases

or decreases from the
maximum, the value of h(x)
decreases. This represents
the height of the projectile
over time.

❑ End behavior?

❑ domain?

❑ range?

The range of f(x) is {y | y ≤ 8}, while
the range of g(x) is all real numbers.

all real numbers.

As x increases or decreases, f(x) decreases.
As x decreases, g(x) decreases, and as x
increases, g(x) increases.

​Complete #8 on your own

Lesson Summary

Stem

❑ I can graph quadratics functions

by…

❑ In analyzing quadratic functions,

I need to…

CHAMPS

INDEPENDENT

Pause & Reflect

❑ Did you struggle

with anything in
this lesson? If so,
how did you deal
with it? What will
you do to
practice the
concepts that you
struggled with?

Forms_Exit_M10_L1

❑Microsoft Forms
❑Find link in

TEAMs, “What
work did I miss”
channel for
today’s date

​Exit Ticket M10 L1 must be turned in before you leave today!! Homework DeltaMath

​Introduction to Parabolas