Analytic Geometry
Unit 5: Quadratic Function
Excerpts from Georgia
Department of Education
Webinar August 7, 2013
melissa.stewart@hallco.org
August 2013
Warm-Up
Which of the following could be the function of a real variable x
whose graph is shown below? Explain.
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August 2013
Answers:
What’s the main idea of Unit 5?
• Interpret the structure of expressions.
• Write expressions in equivalent forms to solve problems
• Create equations that describe numbers or relationships
• Solve equations and inequalities in one variable
• Solve systems of equations
• Interpret functions that arise in applications in terms of the
context
• Analyze functions using different representations
• Build a function that models a relationship between two
quantities
• Build new functions from existing functions
• Construct and compare linear, quadratic, and exponential
models and solve problems
melissa.stewart@hallco.org
August 2013
Concepts & Skills to Maintain from Previous Grades
1. Use Function Notation
2. Put data into tables
3. Graph data from tables
4. Solve one variable linear equations
5. Determine domain of a problem situation
6. Solve for any variable in a multi-variable equation
7. Recognize slope of a linear function as a rate of change
8. Graph linear functions
9. Complex numbers
10. Graph inequalities
Websites to help with the above:
www.aplusmath.com
www.aaamath.com
Enduring Understandings from this Unit
 The graph of any quadratic function is a vertical and/or
horizontal shift of a vertical stretch or shrink of the basic
2
quadratic function f (x) = x .
• The vertex of a quadratic function provides the maximum or
minimum output value of the function and the input at which
it occurs.
• Every quadratic equation can be solved using the Quadratic
Formula.
• The discriminant of a quadratic equation determines whether
the equation has two real roots, one real root, or two
complex conjugate roots.
• Quadratic equations can have complex solutions.
melissa.stewart@hallco.org
August 2013
Examples & Explanations
Problem 1:
The profit that a company makes selling an item (in thousands of
dollars) depends on the price of the item (in dollars). If p is the
price of the item, then three equivalent forms for the profit are:
Which form is most useful in finding:
a.The prices that give a profit of zero dollars? (Factored)
b. The profit when the price is zero? (Standard)
c. The price that gives the maximum profit? (Vertex)
Problem 2:
How many different ways can you think of to solve:
What are some of the different methods?
(Answers vary)
melissa.stewart@hallco.org
August 2013
Problem 3:
Suppose
is an expression giving the
height of a diver above the water (in meters), t seconds after the
diver leaves the springboard.
a. How high above the water is the springboard? Explain how
you know.
When t = 0, the diver is on the board, so, the height is 3
meters.
b. When does the diver hit the water?
seconds
c. When does the diver reach the peak of the dive?
The diver reaches the peak of the dive at the axis of symmetry.
second.
melissa.stewart@hallco.org
August 2013
Problem 4:
Suppose Brett and Andre each throw a baseball into the air. The
height of Brett’s baseball is given by:
Where h is in feet and t is in seconds. The height of Andre’s
baseball is given by the graph:
How long is each baseball airborne?
Brett :
and
seconds
melissa.stewart@hallco.org
August 2013
Andre:
seconds
Problem 5:
A computer game uses functions to simulate the paths of an
archer’s arrow. The x-axis represents the level ground on which
the archer stands, and the coordinate pair (2,5) represents the
top of a castle wall over which he is trying to fire an arrow.
In response to user input, the first arrow followed a path defined
by the function f(x) = 6 - x², failing to clear the castle wall.
The next arrow must be launched with the same force and
trajectory, so the user must reposition the archer in order for his
next arrow to have any chance of clearing the wall.
a. How much closer to the wall must the archer stand in order
for the arrow to clear the wall by the greatest possible
distance?
b. What function must the user enter in order to accomplish
this?
c. If the user can only enter functions of the form f(x + k),
what are all the values of k that would result in the arrow
clearing the castle wall?
This problem can be found at
http://www.illustrativemathematics.org/
melissa.stewart@hallco.org
August 2013
 Websites to assist and enrich:
http://brightstorm.com
http://www.khanacademy.org
 The student edition for Unit 5 can be found at
https://www.georgiastandards.org/CommonCore/Pages/Math-9-12.aspx On the left side, please
look under mathematics, Accelerated Geometry
B/Advanced Algebra. Then, the right side has a pulldown menu to access the units.
 Additional parent guides will be posted to the parent
resource page on
http://www.hallco.org/boe/index.php (right hand
menu) as they become available.
melissa.stewart@hallco.org
August 2013