Algebra I Unit 6

Unit 6: Functions

Algebra I

Section 1: Functions

Vocabulary:

●Function Notation: y=f(x)

●Domain: x values possible for a function

●Range: y values possible for a function

●Ordered pairs: (x,y)

Example 1.

Rewrite the equation, y = 10x, as a function of x.

Example 2.

Rewrite the equation, 5x - 25y + 15 = 0, as a function of x.

Example 3.

Which set of ordered pairs represents the function?

A. {(-3,3), (-2,0), (-1,3), (0,-6)}

B. {(-3,3), (-2,4), (-1,5), (0,6)}

C. {(-3,-3), (-2,0), (-1,3), (0,6)}

D. {(-3,-3), (-2,-4), (-1,-5), (0,-6)}

Example 4.

Given the function below, find f(t).

Example 5.

Given the function below, find f(m - n).

Example 6.

The function, T(h), represents the temperature, in degrees Fahrenheit, h hours
after 6 p.m.

What does the statement, T(0) = 58 represent?

Example 7.

The function, P(y), represents a company's yearly profit, in thousands of
dollars, y years after 1995.

What does the statement, P(30) - P(25) = 1,450, represent?

Section 2: Interpreting Functions

Example 1.

Pete's Pizzeria charges a $2.50 delivery fee for all deliveries and $5.99 for each
medium pizza ordered, tax included. The following equation represents the
total cost, y, for a customer who orders x medium pizzas and has it delivered.

What does the y-intercept represent?

Example 2.

The linear model below shows the population of a town x years after 1980.

According to the model, what was the population in 1980?

Example 3.

Maria is attending County Community College in the fall. Each semester, the
community college charges students $275.00 per class, plus $325.00 in fees.
The function below represents Maria's semester cost, C(x), given that she
enrolls in x classes.

Suppose Maria draws a graph to represent her semester cost. Determine
whether the graph is increasing or decreasing.

Example 4.

Juan received $200.00 from his grandmother for his birthday, so he decides to
open a non-interest-bearing savings account. Each month, he plans to deposit
$8.00 into the account. The function below represents the balance of Juan's
account, A(x), after x months.

Suppose Juan draws a graph to represent the balance of his account. Describe
the end behavior of the graph.

Example 5.

Anita won $500.00 at a cooking competition, so she decides to open an
interest-bearing savings account, which is compounded continuously at a
fixed annual interest rate of 3.5%. The function below represents the balance
of Anita's account, A(x), after x months.

What does the y-intercept represent?

Example 6.

Jamie is purchasing a new car. She knows that the value of the car will
decrease by 12% each year from the date of purchase. The function below
represents the value of the car, V(x), x years after the date of purchase.

Suppose Jamie draws a graph to represent the value of the car. Determine
whether the graph is increasing or decreasing, and describe the end behavior.

Example 7.

A college student launches a new social networking website. The number of users is
increasing at a rate of 27% per month from the website's launch. The function below
represents the number of users, N(x), x months after the website's launch.

The college student wants to examine the number of users on the website from 3 months
after the website's launch to 6 months after the website's launch. What is the minimum
number of users on this interval?

Example 8.

A new telecommunications company is tracking the number of cell phone subscribers in a small city each month.
The initial number of cell phone subscribers in the city was 1,540. The table below represents the number of cell
phone subscribers x months after the company began its operations.

Suppose they draw a graph to represent the number of cell phone subscribers. Determine whether the graph is
increasing or decreasing, and describe the end behavior.

Months, x

1

2

3

4

5

Number of Subscribers, N(x)

1540

3080

6160

12320

49280

Example 9.

A launcher throws a ball from the top of a building at an height of 48 feet. The ball reaches a
maximum height of 98 feet after a time of 10 seconds. The ball falls to the ground after 24
seconds. This can be modeled by the quadratic function given below.

What is the rate of change from the original position until the ball reaches its maximum
height?

Example 10.

P(x) is a profit function where x represents the sales price of an item.

Find the profit made when the item is priced at $20.

Section 3: Domain and Range

Example 1.

Determine the domain and range of the linear function, f(x) = x, shown in the graph below.

Example 2.

Determine the domain and range of the function below.

Example 3.

A local youth group is planning a trip to an amusement park. They are taking a bus
which can hold 32 people. The mandatory parking cost is $25 even if no tickets are
purchased, and tickets to the park cost $22.50 per person. The following function gives
the cost of visiting the amusement park, c, and the number of people who purchase the
ticket, n.

Rules of Domain and Range:

●Domain:

○Quadratic Function: -∞ < x < ∞

○Exponential Function: -∞ < x < ∞

●Range:

○Quadratic Function: If it opens upward min < x < ∞, If it opens downward -∞ < x < max

○Exponential Equation: positive function- never has an output less than or equal to the
asymptote, negative function- never has an output greater than the asymptote

Example 4.

What are the domain and range of the function graphed below?

Example 5.

What are the domain and range of the function below?

f(x) = (x + 15)2 - 24

Example 6.

Find the domain and range of the function graphed below.

Example 7.

Find the domain and range of the function below.

Section 4: Rate of Change

Vocabulary:

Rate of Change: slope of an equation

Formula for rate of change:

Example 1.

Paula has to read a novel for her English class. The graph below represents
the number of pages she has left to read after x hours of reading. What is the
rate of change of the graph?

Example 2.

Melissa is feeding the ducks at the pond. She is going through her loaf of bread at a constant rate.
The table below shows the number of slices of bread remaining in relation to the number of minutes
she has been feeding the ducks.

Minutes

1

2

3

Slices of Bread

33

31

29

Example 3.

Miguel, a photographer, uses a linear model to show the relationship between
the number of photos sold and the amount of money, in dollars, he earned.
Interpret the meaning of the rate of change.

Vocabulary:

●Average rate of change: ratio of change in y to x

Formula for Average Rate of Change:

Example 4.

Find the average rate of change of the function below over the interval [0, 2].

Example 5.

Find the average rate of change of the following function over the interval
[3,7].

Example 6.

The table represents the function h(x). Find the approximate average rate of change of h(x)
over the interval [-6, 0].

x

-6

-4

-2

0

2

h(x)

0.125

0.25

0.5

1

2

Section 5: Linear and Exponential Functions

Slope Intercept Form:

y=mx+b

Example 1.

Which graph above shows a line with a slope of and a y-intercept of 3?

Example 2.

Which graph represents the equation above?

Example 3.

What are the x-intercept and the y-intercept of the line from the equation
below?

Example 4.

What are the x-intercept and the y-intercept of the line from the equation
below?

Exponential Functions:

f(x)=bx, where b>0 and b≠ 1

Rules:

1.

Horizontal asymptote lies on or above the x axis means there is no x
intercept

2.

Horizontal asymptote lies below the x axis then there is an x intercept

Example 5.

What is the x-intercept of the following exponential function?

Example 6.

What is the x-intercept of the following exponential function?

Example 7.

What is the y-intercept of the following function?

Example 8.

Graph the following exponential function.

Example 9.

Graph the function, and choose the key features that apply to the graph.

Example 10.

Bella bought some school supplies from a store. She bought a backpack and some pencils. The
following equation represents the total amount in dollars, $y, spent by Bella in buying x pencils and a
backpack. What does the y-intercept represent?

Example 11.

Jade is attending an art school that charges its students per class along with a
one-time registration fee in dollars. The function below represents Jade's cost,
$A(x), given that she enrolls in x classes. What does the slope of the function
represent?

Example 12.

Charlie bought a rare vintage coin from an auction. She knows that the value of the coin will increase
each year from the date of purchase. The function below represents the value of the coin, $C(x), x
years after the purchase. What does the y-intercept in the above function represent?

Section 6: Quadratic Functions

Vocabulary:

●Extreme Value: when a quadratic function is in vertex form the value of k,
which is equal to f(h).

●Minimum: If the quadratic function opens upward, the lowest point

●Maximum: If the quadratic function opens downward, the highest point

●Zeros: where the function crosses the x axis

Three forms of quadratic functions:

1.

Standard form, ax2+bx+c

2.

Factored form, a(x-b)(x-c)

3.

Vertex form, f(x)=a(x-h)2+k

Example 1.

Graph the following function.

Example 2.

Find the maximum or minimum of the parabola given by the function below.

Example 3.

Find the extreme value of the following quadratic function by completing the
square.

Example 4.

Find the zeros of the following function.

Example 5.

Find the zeros of the following function by completing the square.

Example 6.

Identify the function that fits the situation and solve the problem.

A baseball is thrown up in the air with an initial velocity of 30 feet per second
from an initial height of 4 feet, with an acceleration of 32 feet per second
squared due to gravity. How long is the baseball in the air?

Section 7: Compare Properties of Functions

Example 1.

A linear function, f(x), contains the points (-2, 46) and (6, -18).

If g(x) = 12x - 25, which statement is true?

A.The graphs of f(x) and g(x) have the same steepness with negative slopes.

B.The graphs of f(x) and g(x) have the same steepness with positive slopes.

C.The graph of g(x) is less steep than the graph of f(x).

D.The graph of g(x) is steeper than the graph of f(x).

Example 2.

The graph of f(x) is shown below.

If g(x) = 2(x - 2)2 + 3, which statement is true?

A.The minimum value of f(x) is less than the minimum value of g(x).

B.The minimum value of f(x) is greater than the minimum value of g(x).

C.The minimum value of f(x) is equal to the minimum value of g(x).

Example 3.

The graph below represents the function f(x), and the table below represents the function g(x).
Which statement is true?

A.The functions f(x) and g(x) both intersect the x-axis at (-2, 0).

B.The functions f(x) and g(x) never intersect the x-axis.

C.The function f(x) intersects the x-axis at (-2, 0), but the function g(x) never intersects the x-axis.

D.The function f(x) intersects the x-axis at (-2, 0), but the function g(x) intersects the x-axis at (-3, 0).