Transcript 1.2 Functions

§ 1-2 Functions
The student will learn about:
functions, domain, range,
linear functions,
quadratic functions,
functions used in a
business setting,
and some new
calculator techniques.
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Definition of a Function
Def. – a function f is a rule that assigns to each
number x in a set a number f (x). That is, for
each element in the first set there corresponds
one and only one element in the second set.
That’s one – not two, not three, but one, 1, uno,
un, eine, no more no less, but exactly one.
Example – For the sale of x items there
corresponds a revenue.
The first set is called the domain, and the set of
corresponding elements in the second set is called
the range.
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Functions as a Machine
A function f may be thought of as a numerical
procedure or “machine” that takes an “input”
number x and produces an “output” number f(x),
as shown below. The permissible input numbers
form the domain, and the resulting output
numbers form the range.
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Functions Defined by Equations
2x – y = 1
x2 + y2 = 25 (Not a function – graph it.)
Note if x = 3 then y = both 4 and – 4.
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USING THE VERTICLE LINE TEST
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Functions
• The domain and range can be illustrated
graphically.
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Functions Defined by Equations
Finding the domain:
1. Eliminate square roots of negatives
f (x)  x  3
This implies that x – 3 ≥ 0 or x ≥ 3.
2. Eliminate division by zero
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f (x) 
x5
This implies that x – 5 ≠ 0 or x ≠ 5.
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Introduction to Linear Functions
A straight line has an equation of the
form Ax + By = C.
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Linear Functions And Equations.
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Cost Function
Finding a company’s cost function
Cost Function
C (x) = mx + b
C is the total cost and
mx is the variable cost.
b (the y-intercept) is the fixed cost.
m (slope) is the unit cost.
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Example
A company manufactures bicycles at a cost of
$55 each. If the fixed cost are $900, express the
company’s cost as a linear function.
Cost Function
C (x) = mx + b
Where m (slope) is the unit cost and b (the y-intercept) is the
fixed cost.
The unit cost is $55 and the fixed cost is $900 hence
C (x) = 55 x + 900
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Example
Cost
The graph of C (x) = 55 x + 900 is a line with slope 55
and y-intercept 900, as shown below. The slope is the unit
cost which is the same as the rate of change of the cost,
which is also the company’s marginal cost. The yintercept is the fixed cost.
C (x) = 55 x + 900
y-intercept 900
units
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Introduction to Quadratic Functions
A quadratic is an equation of the form
f (x) = a x2 + b x + c , and graphs as a parabola.
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Definition
NOTE:
a is the coefficient of the x 2 term,
b is the coefficient of the x term, and
c is the constant term.
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Functions
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Fact of Interest
A quadratic function graph can either open up
or open down. If a > 0 the graph opens up and
if a < 0 the graph opens down.
f (x) =
+ x – 2.
a=1
x2
f (x) = - 2x 2 + 4x + 6
a=-2
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Functions
The vertex of a parabola is its “central” point,
the lowest point on the parabola if it opens up
and the highest point if it opens down.
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Definition
The vertex (lowest or highest point) is of
importance and the x value can be found as
follows:
BUT - there is an easier way!
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Graphing a Quadratic
Function: Calculator
1. Turn the calculator on and press the y =
button. If something is there press clear.
2. Enter the function Y1 = x 2 + x – 2 and then
press ZOOM and 6 for a standard window.
3. Find the x and y intercepts using the calculator.
x-intercepts using CALC
and zero giving 1 and -2
y-intercept using CALC
and value giving -2
OR use table!
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Graphing (Continued)
4. The vertex of the parabola can be found on
the calculator using the minimum or maximum
menu options.
5. Press CALC and 3 for a minimum.
Giving x = - 0.5 and
y = x2 + x – 2
y = - 2.25
NOTE: There is no need
to use the previous
formula for the x-value.
b 1
1
x


2a
2
2
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       I love my calculator!       
Solving a Quadratic Equation
f (x) = x2 + x – 2.
1. The solutions of a quadratic (the x-intercepts)
are also called roots or zeros. We just found the
x-intercepts using the calculator.
2. Solutions may also be found algebraically.
Let f (x) = 0 and solve for x :
a. Factor : 0 = (x + 2)(x – 1) and x = 1 and – 2.
b. Complete the square – No thank you!
c. Use the quadratic formula – Next slide please!
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Solving a Quadratic Equation
f (x) = x2 + x – 2.
c. Use the quadratic formula :
a = 1, b = 1, c = -2.
1 1 8 1  3
x

 1,  2
2
2
OR use your calculator under calc and zeros.
       I love my calculator!      22
Quadratic Review
y = ax 2 + bx + c
x intercepts
CALC & ZERO
(x, 0)
y intercept
CALC & VALUE
(0, y)
opens up if
a>0
opens down if
a<0
vertex at
CALC & MAX or MIN
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Basic Business Applications.
Cost Function
C = Unit cost · Quantity + Fixed cost
Where m (slope) is the unit cost and b (the
y-intercept) is the fixed cost.
Revenue Function
R = (price) · (quantity sold)
(Note: quantity is number of items sold at price
of $p.)
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Basic Business Applications.
Profit Function
P=R–C
Let C represent cost, R represent revenue and
P profit. Then one of three things can occur:
R>C
P>0
a profit
R=C
P=0
a break-even point, or
R<C
P<0
a loss.
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Application Example
Break-even Analysis.
Use the revenue and cost functions,
R (x) = x (2,000 – 60x) & C (x) = 4,000 + 500 x, where x
is thousands of dollars. Both functions have domain 0 ≤ x ≤ 25.
1. Sketch the graph of both functions on the
same coordinate system. (Algebraically)
Do you have any idea how long this would
take or how difficult it is to do?
2. Find the break-even point
(R (x) = C (x)) (Algebraically).
Do you have any idea how long this would take or
how difficult it is to do? Well do you? 26
Application Example by Calculator
Break-even Analysis.
Use the revenue and cost functions,
R (x) = x (2,000 – 60x) & C (x) = 4,000 + 500 x,
where x is thousands of dollars. Both functions have
domain 0 ≤ x ≤ 25.
0 ≤ x ≤ 25
0 ≤ y ≤ 17,000
1. Graph both functions on your calculator.
2. Use Calc & intersect
to find the break-even
point.
3. When does profit
occur?
(3.03, 5518)
(21.96, 14982)
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       I love my calculator!       
Solving Quadratic Equations
The relationship among the cost, revenue, and
profit functions can be seen graphically as
follows.
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Application Example by Calculator
Break-even Analysis. This is the problem we just did!
Use the revenue and cost functions,
R (x) = x (2,000 – 60x) & C (x) = 4,000 + 500 x,
where x is thousands of dollars. Both functions have
domain 0 ≤ x ≤ 25.
For homework, graph the
profit equation with these
two graphs – revenue and
cost.
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      Embrace your calculator!     

Number of
units
Revenue
Cost
Profit
x
R(x)
C(x)
P(x)
0
0
4000
-4000
1
1940
4500
-2560
2
3760
5000
-1240
3
5460
5500
-40
4
7040
6000
1040
5
8500
6500
2000
6
9840
7000
2840
7
11060
7500
3560
8
12160
8000
4160
9
13140
8500
4640
10
14000
9000
5000
11
14740
9500
5240
12
15360
10000
5360
13
15860
10500
5360
14
16240
11000
5240
15
16500
11500
5000
16
16640
12000
4640
17
16660
12500
4160
18
16560
13000
3560
19
16340
13500
2840
20
16000
14000
2000
21
15540
14500
1040
22
14960
15000
-40
23
14260
15500
-1240
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13440
16000
-2560
P
r
o
f
i
t
Spreadsheet
Exploration
A spreadsheet may be substituted
for a graphing calculator.
20000
15000
10000
R(x)
C(x)
5000
P(x)
0
0
5
10
15
20
25
-5000
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Summary.
• We learned about functions and the basic terms
involved with functions.
• We learned about the linear functions.
• We learned about the quadratic functions.
• We learned about the basic business functions.
• We learned how to use a graphing calculator to
make our work much easier.
• We saw how spreadsheets can replace the work of
a graphing calculator.
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       I love my calculator!       
ASSIGNMENT
§1.2 on my website
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