Download ACT – Class Opener: Recall: Polynomial Function

ACT – Class Opener:
• If a circle’s area is 16𝜋 square centimeters, what
is the length, in centimeters of its diameter?
• Troy works as a server at a restaurant. At the end
1
of the week he give of his total tips to the
4
1
gives
4
hostess. After that, he
of his remaining tip
money to the cooks and they split it evenly. If
each cook receives $6, how much money did Troy
start out with?
Recall: Polynomial Function
• A polynomial function is a function that can
be defined by evaluating a polynomial. A
function ƒ of one argument is called a
polynomial function if it satisfies for all
arguments x, where n is a non-negative
integer and a0, a1,a2, ..., an are constant
coefficients.
Degrees of Polynomial Functions
• Constant function such as f(x)=a has a degree
of zero.
• Linear functions, f(x) = mx+b, have a degree of
one.
• Quadratic functions have a degree of two.
Quadratic Functions:
• Let a, b, and c be real numbers with a ≠0. The
function given by
𝑓 𝑥 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
is called a quadratic function.
Quadratic Functions
• All quadratic Functions will produce:
– Parabolas
– Axis of Symmetry
– Vertex
Example 1:
• Describe how the graph of each function is
related to the graph of 𝑦 = 𝑥 2 .
1 2
𝑓 𝑥 = 𝑥
3
𝑓 𝑥 = 2𝑥 2
𝑘 𝑥 = 𝑥+2
2
−3
Standard Form of a Quadratic
Function
• 𝑓 𝑥 =𝑎 𝑥−ℎ
2
+ 𝑘,
𝑎≠0
• The graph of f is a parabola whose axis is the
vertical line x = h and whose vertex is the
point (h,k). If a > 0, the parabola opens
upward, if a < 0 the parabola opens
downward.
Example 2:
• Describe the graph of the following function
and identify the vertex.
𝑓 𝑥 = 2𝑥 2 + 8𝑥 + 7
• Write the function in standard form.
Student Check:
• Rewrite each quadratic function in standard
form and identify the vertex:
𝑥 2 − 10𝑥 + 25
𝑥 2 − 8𝑥 + 16
4𝑥 2 − 4𝑥 + 21
Identifying the X intercepts of a
Quadratic Equation
• Find the x intercepts of the following
quadratic equation:
𝒇 𝒙 = −𝒙𝟐 + 𝟔𝒙 − 𝟖
• Rewrite the quadratic equation in standard
form.
Student Check
• Find the x intercepts of the following
quadratic equations:
𝒇 𝒙 = −𝒙𝟐 + 𝟐𝒙 + 𝟓
𝟓
𝒇 𝒙 =𝒙 −𝒙+
𝟒
𝟐
𝒇 𝒙 = 𝒙𝟐 + 𝟐𝒙 + 𝟏
ACT Class Opener:
• Mr. Mauro gave his
class a test on 25
vocabulary words.
Only one of the
following percent is
possible as the percent
of 25 words a student
defined correctly.
Which one is it.
A) 99%
B) 80%
C) 69%
D) 45%
E) 26%
Writing in Standard Form
• Write the standard form of the equation of
the parabola whose vertex is (1,2) and that
passes through the points (3,-6)
Student Check:
• Write the equation of a parabola with vertex
(-2,5) and passes through the point (0,9)
Finding Minimum and Maximum
• If a > 0 f has a minimum value at 𝑥 =
𝑏
−
2𝑎
• If a < 0 f has a maximum value at 𝑥 =
𝑏
−
2𝑎
The Maximum Height of a Baseball
• A baseball is hit at a point 3 ft above the
ground at a velocity of 100 ft/sec and at an
angle of 45 degrees with respect to the
ground. The path of the baseball is given by
the function 𝒇 𝒙 = −. 𝟎𝟎𝟑𝟐𝒙𝟐 + 𝒙 + 𝟑
where f(x) is the height of the baseball and x is
the horizontal distance from home plate.
What is the maximum height the ball reaches?
Example:
• A soft drink company has daily production
costs of:
𝐶 𝑥 = 70,000 − 120𝑥 + .055𝑥 2
where C is the total cost and x is the number
of units produced. Estimate numerically the
number of units that should be produced each
day to yield a minimum cost.
Quick Quiz:
• The number, g, of grants awarded from the
Nation Endowment for Humanities fund from
1999 to 2003 can be approximated by the model:
𝑓 𝑥 = −99.14𝑥 2 + 2201.1𝑥 − 10896
where x represents the year, with x = 9
corresponding to 1999. Using this model determine
the year in which the number of grants awarded
was greatest.
Partner Practice:
• Complete the following:
Pg. 99 – 102
#21 – 26
#29 – 34
#55 – 61