Outline
Graphs of Quadratic Functions
Application
College Algebra & Trigonometry I
3.1 - Quadratic Functions
Math 1100
North Carolina Central University
Math & C.S. Department
Hicham Qasmi - hqasmi@nccu.edu
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
1
Graphs of Quadratic Functions
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
2
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
1
Graphs of Quadratic Functions
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
2
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Outline
1
Graphs of Quadratic Functions
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
2
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Graph of Quadratic Functions
Definition
A quadratic function in x is a function f (x) of the form
f (x) = ax 2 + bx + c
where a, b, and c are real numbers with a 6= 0.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Graph of Quadratic Functions
Definition
A quadratic function in x is a function f (x) of the form
f (x) = ax 2 + bx + c
where a, b, and c are real numbers with a 6= 0.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Example of a Quadratic Function with a > 0
The graph of a quadratic function is called a parabola:
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Example of a Quadratic Function with a < 0
The graph of a quadratic function with a < 0:
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Standard Form
Theorem
A quadratic function f is in its standard form if f (x) has the
form
f (x) = a(x − h)2 + k with a 6= 0
The graph of f is a parabola whose vertex is the point (h, k)
The parabola is symmetric with respect to the line x = h.
If a > 0, the parabola opens upward.
If a < 0, the parabola opens downward.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Standard Form
Theorem
A quadratic function f is in its standard form if f (x) has the
form
f (x) = a(x − h)2 + k with a 6= 0
The graph of f is a parabola whose vertex is the point (h, k)
The parabola is symmetric with respect to the line x = h.
If a > 0, the parabola opens upward.
If a < 0, the parabola opens downward.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Standard Form
Theorem
A quadratic function f is in its standard form if f (x) has the
form
f (x) = a(x − h)2 + k with a 6= 0
The graph of f is a parabola whose vertex is the point (h, k)
The parabola is symmetric with respect to the line x = h.
If a > 0, the parabola opens upward.
If a < 0, the parabola opens downward.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Standard Form
Theorem
A quadratic function f is in its standard form if f (x) has the
form
f (x) = a(x − h)2 + k with a 6= 0
The graph of f is a parabola whose vertex is the point (h, k)
The parabola is symmetric with respect to the line x = h.
If a > 0, the parabola opens upward.
If a < 0, the parabola opens downward.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Standard Form
Theorem
A quadratic function f is in its standard form if f (x) has the
form
f (x) = a(x − h)2 + k with a 6= 0
The graph of f is a parabola whose vertex is the point (h, k)
The parabola is symmetric with respect to the line x = h.
If a > 0, the parabola opens upward.
If a < 0, the parabola opens downward.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Standard Form
Theorem
A quadratic function f is in its standard form if f (x) has the
form
f (x) = a(x − h)2 + k with a 6= 0
The graph of f is a parabola whose vertex is the point (h, k)
The parabola is symmetric with respect to the line x = h.
If a > 0, the parabola opens upward.
If a < 0, the parabola opens downward.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Outline
1
Graphs of Quadratic Functions
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
2
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Graphing in Standard Form
How to graph f (x) = a(x − h)2 + k?
Determine whether the parabola opens upward or
downward using the sign of a.
Determine the vertex (h, k) of the parabola.
Find any x-intercepts by solving f (x) = 0.
Find the y-intercept by computing f (0).
Plot the intercepts, vertex, and any other points you need.
Connect the points using a smooth curve.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Graphing in Standard Form
How to graph f (x) = a(x − h)2 + k?
Determine whether the parabola opens upward or
downward using the sign of a.
Determine the vertex (h, k) of the parabola.
Find any x-intercepts by solving f (x) = 0.
Find the y-intercept by computing f (0).
Plot the intercepts, vertex, and any other points you need.
Connect the points using a smooth curve.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Graphing in Standard Form
How to graph f (x) = a(x − h)2 + k?
Determine whether the parabola opens upward or
downward using the sign of a.
Determine the vertex (h, k) of the parabola.
Find any x-intercepts by solving f (x) = 0.
Find the y-intercept by computing f (0).
Plot the intercepts, vertex, and any other points you need.
Connect the points using a smooth curve.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Graphing in Standard Form
How to graph f (x) = a(x − h)2 + k?
Determine whether the parabola opens upward or
downward using the sign of a.
Determine the vertex (h, k) of the parabola.
Find any x-intercepts by solving f (x) = 0.
Find the y-intercept by computing f (0).
Plot the intercepts, vertex, and any other points you need.
Connect the points using a smooth curve.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Graphing in Standard Form
How to graph f (x) = a(x − h)2 + k?
Determine whether the parabola opens upward or
downward using the sign of a.
Determine the vertex (h, k) of the parabola.
Find any x-intercepts by solving f (x) = 0.
Find the y-intercept by computing f (0).
Plot the intercepts, vertex, and any other points you need.
Connect the points using a smooth curve.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Graphing in Standard Form
How to graph f (x) = a(x − h)2 + k?
Determine whether the parabola opens upward or
downward using the sign of a.
Determine the vertex (h, k) of the parabola.
Find any x-intercepts by solving f (x) = 0.
Find the y-intercept by computing f (0).
Plot the intercepts, vertex, and any other points you need.
Connect the points using a smooth curve.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Graphing Example 1
Example
Graph
f (x) = −2(x − 3)2 + 8
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Graphing Example 2
Example
Graph
f (x) = (x + 3)2 + 1
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Outline
1
Graphs of Quadratic Functions
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
2
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Graphing f (x) = ax 2 + bx + c
Theorem
The coordinates vertex of the parabola defined by the quadratic
function f (x) = ax 2 + bx + c are
!
b b −
;f −
2a
2a
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Graphing Example 1
Example
Graph
f (x) = −x 2 − 2x + 1
. Use the graph to identify the functions domain and range.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Minimum/Maximum Values
Theorem
Consider the quadratic function f (x) = ax 2 + bx + c.
b
.
If a > 0, then f has a minimum that occurs when x = − 2a
b
The minimum value of f is f − 2a .
If a < 0, then f has a maximum that occurs when
b
.
x = − 2a
b
The maximum value of f is f − 2a
.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Minimum/Maximum Values
Theorem
Consider the quadratic function f (x) = ax 2 + bx + c.
b
.
If a > 0, then f has a minimum that occurs when x = − 2a
b
The minimum value of f is f − 2a .
If a < 0, then f has a maximum that occurs when
b
.
x = − 2a
b
The maximum value of f is f − 2a
.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Minimum/Maximum Values
Theorem
Consider the quadratic function f (x) = ax 2 + bx + c.
b
.
If a > 0, then f has a minimum that occurs when x = − 2a
b
The minimum value of f is f − 2a .
If a < 0, then f has a maximum that occurs when
b
.
x = − 2a
b
The maximum value of f is f − 2a
.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Minimum/Maximum Values
Theorem
Consider the quadratic function f (x) = ax 2 + bx + c.
b
.
If a > 0, then f has a minimum that occurs when x = − 2a
b
The minimum value of f is f − 2a .
If a < 0, then f has a maximum that occurs when
b
.
x = − 2a
b
The maximum value of f is f − 2a
.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Minimum/Maximum Values
Theorem
Consider the quadratic function f (x) = ax 2 + bx + c.
b
.
If a > 0, then f has a minimum that occurs when x = − 2a
b
The minimum value of f is f − 2a .
If a < 0, then f has a maximum that occurs when
b
.
x = − 2a
b
The maximum value of f is f − 2a
.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
Min/Max Example
Example
Consider
f (x) = −3x 2 + 6x − 13
.
1
Determine without graphing whether the function has a
minimum value or maximum value.
2
Find the minimum or maximum value and determine where
it occurs.
3
Identify the functions domain and range.
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Outline
1
Graphs of Quadratic Functions
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
2
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Application 1: Food Stamp Program
Application
The function
f (x) = 0.22x 2 − 0.50x + 7.68
models the number of households, f (x), in millions,
participating in the food stamp program x years after 1999.
1
According to this function, in what year was the number of
participants minimum?
2
How many households received food stamps that year?
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Outline
1
Graphs of Quadratic Functions
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
2
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Application 2: Pairs of Numbers
Application
Among all pairs of numbers whose difference is 10,
1
find a pair whose product is as small as possible.
2
What is the minimum product?
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Outline
1
Graphs of Quadratic Functions
Parabolas
Graphing Quadratic Functions in Standard Form
Graphing f (x) = ax 2 + bx + c
2
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Math 1100
College Algebra & Trigonometry I
Outline
Graphs of Quadratic Functions
Application
Households Food Stamp Program
Pairs of Numbers
Fencing and Optimal Area
Application 3: Fencing and Optimal Area
Application
You have 100 yard of fencing to enclose a rectangular area.
1
Find the dimensions of the rectangle that maximizes the
enclosed area.
2
What is the maximum area?
Math 1100
College Algebra & Trigonometry I