Section 1.4 Linear Functions and Models NOTES
Precalculus
Previously, we have learned that a relation between two values, x and y, is a
function if for every x value there is exactly one y value.
y
y
A
x
x
B
Using this analysis:
Is the graph of A above a function? ______________
Is the graph of B above a function? ______________
A function is often denoted by the notation f(x). In words, what does f(x) say?
___________________________________
When you see something such as f(x) = 3x + 5, the f(x) portion is essentially the
same as y, so f(x) = 3x + 5 means about the same as y = 3x + 5. The x in f(x) tells
us what to replace x with in the equation. For example, for f(x) = 3x + 5,
f(a) = 3a + 5, and f(7) = 3 x 7 + 5 = 21 + 5 = 26.
Example: If f(x) = 3x2 – 5, what is the value of f(7)? _______________________
A value of x that makes a function equal to zero is called the zero of a function.
Example: What is the zero of f(x) = 2x + 4?
The zero is ________
What is the form for linear functions? _______________________________
Use Example 1 to do the following problem:
A club is putting on a meal to make some money. It will cost $400 for the food,
and tickets for the meal are being sold for $10 each.
a.) Express the net income, f(x), as a function of the number of tickets, t,
sold.
b.) Graph this function on the graph below from t = 0 to t = 100.
c.) What is the break-even for this situation?
The domain of a function is the values of x allowed by the function and the range is
the values of y (f(x)) allowed by the function.
For the function f  x   x  3
What is the domain for x? _______________________________________
What is the range for f(x)? ______________________________________
Using Example 2, do the following problem:
It costs $1.40 for the first minute of a phone call to Paris, France, and $0.80 for
each additional minute or fraction thereof.
Draw the graph of a step function that models this cost.
Section 1.4 Linear Functions and Models ASSIGNMENT
Precalculus
1.) If f  x  
3
1
x  , find f(2) and f(-2). Then find the zero of f.
4
2
f(2) =
f(-2) =
The zero of f is ___________
2.) If C(n) = 20 –
5
n , find C(0) and C(16). Then find the zero of C.
8
C(0) =
C(16) =
The zero of C is ___________
3.) Let f(x) = 3x – 7 . Decide whether f(2) + f(6) = f(8)?
Is this true or false? _______________
4.) Consider the constant function P(x) = -0.5.
a.) Find P(1269.35)
P(1269.35) = ___________________
b.) Does the function P have any zeros? __________
5.) a.) What is the slope of the graph of f(x) = 1.5x – 2?
b.) What is the zero of f(x)?
c.) What is the x-axis intercept of f(x)?
___________
6.) Let f be a linear function such that f(1) = 5 and f(3) = 9.
a.) Sketch the graph of f.
b.) Find an equation for f(x).
7.) Maria Correia’s new car costs $280 per month for car payments and insurance.
She estimates that gas and maintenance cost $0.15 per mile.
a.) Express her total monthly cost, C, as a function of the miles, m, driven
during the month.
b.) what is the slope of the graph of the cost function?
8.) A recording studio invests $24,000 to produce a master tape of a singing
group. It costs $1.50 to make each copy of the master and cover the operating
costs.
a.) Express the cost of producing t tapes as a function C(t).
b.) If each tape is sold for $6.50, express the revenue (the total amount
received from the sale) as a function R(t).
c.) Find the coordinates of the break-even point (that is, where the graphs
intersect).
9.) At a city garage, it costs $4 to park for the first hour and $2 for each additional
hour or fraction thereof. The fee is a function of the time parked. Sketch the
graph of a function that models this parking fee.
10.) A businesswoman buys a $12,000 computer system. The value of the system
is considered to depreciate by a fixed amount, which is 10% of the purchase
price each year over a ten-year period.
a.) Express the value of the computer system as a function of the number of
years since its purchase.
b.) What is the domain of the function (in determining this, assume that the
value of the computer system will never have a negative dollar value).
11.) The last test that Mr. Clements gave was so hard that he decided to scale the
grades upward. He decided to raise the lowest score of 47 to a 65 and the
highest score of 78 to a 90. Find a linear function that would give a fair way to
convert the other test scores.