AST: Lesson 2.1 – The Language of Functions
Relation – a pairing of input values with output values
Independent variable –
the value that you control
or manipulate; the input
(often denoted by x)
Dependent variable –
depends in some way on
the independent variable,
the output
(often denoted by y)
Function – a relation such that each domain is paired with exactly one range
Domain – the set of all inputs (usually represented by the x-variable)
Range – the set of all outputs (usually represented by the y-variable)
Understanding a Function (Color Examples)
Functions
Not a Function
Example 1: the cost of renting a car is $35 a day plus $0.40/mile for mileage over 100 miles.
a. Which statement is true…
…”cost is a function of miles driven”
OR …”miles driven is a function of cost”
b. Identify the independent and dependent
function.
variables.
c. State the domain and range of the
Independent 
Domain 
Dependent 
Range 
Function Notation (Euler’s Notation) – indicates the value of the range when given “x”
Symbol:
f(x)
Read:
“f of x”
Means:
f(x) depends on the value of x
Example 2: Let f(x) = 3x2 – 1
a. Evaluate f(1), f(4) and f(5).
b. Does f(1) + f(4) = f(5)?
Vertical line test –
the method of
drawing vertical
lines to determine if
the graph of a
relation is a function
Example 3 – Determine whether each relation is a function using the vertical line test.
ALSO, identify the domain and range of each.
AST: Lesson 2.2 – Linear Models
Linear function –
 a function that is represented by a constant rate of change between the two variables
 graph is a straight line
 modeled by the equation y = mx + b
o m is the slope (rate of change)
o b is the y-intercept (initial amount)
Example 1 – A 140-lb. woman diets and loses 0.5 pounds per week. Let y be her weight after w
weeks, where 0 < w < 20.
a. Find an equation that expresses y in terms of w.
b. At this rate of weight loss,
what will her weight be in 5 weeks?
c. At this rate of weight loss, when will
she reach her goal weight of 132 lbs?
Example 2 – The points (1.5, -.565), (2, -.89) and (3.5, -1.865) lie on a line. Write the equation of
the line in slope-intercept form.
Some functions may be “almost” linear, but not exactly. In these situations, we create a scatter-plot
and draw a line-of-best-fit.
Scatter Plot Correlations & Line-of-Best-Fit
Example 3 – A biologist counted the number of chirps per minute by crickets at various temperatures
over a period of time, and obtained the following data.
Temperature (℉)
No. of Chirps (per min)
52
61
53
63
56
76
58
88
59
92
67
125
73
145
77
161
a. Graph the points from the table onto the coordinate plane.
Be sure to label the axes.
b. Find the model to represent the relationship between
temperature and number of chirps.
c. Use your model to find the number of chirps you would expect at each of the following
temperatures.
i. 72 ℉
ii. 85 ℉
Interpolation – the process by which data is estimated and falls between known points on the line
Extrapolation – the process by which data is estimated and lies outside either of the known
endpoints
AST: Lesson 2.3 – The Line of Best Fit
Lesson 2.2 - Example 3: A biologist counted the number of chirps per minute by crickets at various
temperatures over a period of time, and obtained the following data.
Temperature (℉)
No. of Chirps (per min)
52
61
53
63
56
76
58
88
59
92
67
125
73
145
77
161
Observed Values – data collected from sources such as experiments and surveys
Predicted Values – data calculated using the predicted linear model (i.e. equation of line-of-best-fit)
Linear Model for Example 3: ________________________
(To calculate the predicted value of chirps, substitute each temperature into the equation above)
Temperature
(x)
No. of
Chirps
(Observed)
52
61
53
63
56
76
58
88
59
92
67
125
73
145
77
161
No. of
Chirps
(Predicted)
Error
(Observed –
Predicted)
Squares of
the Errors
Sum of Squares
Errors (Deviations) – the differences between the observed and expected values; the best line-of-fit
passes as close as possible to all observed data
(To find the error between actual & predicted values, subtract observed chirps – predicted chirps)
Method of Least Squares – the process of finding the best line-of-fit by comparing the “sum of
squares”
The ______________ the sum of squares, the ________________ the line of fit.
Center of Gravity – a point that should lie on the line of fit
∑

Find the mean of x-coordinates  𝑥̅ =

Find the mean of observed y-coordinates  𝑦 =

Put the means together as an ordered pair (𝑥̅ , 𝑦)
∑
a. Calculate the Center of Gravity for the previous example.
b. Check the Line of Fit using the Center of Gravity.
Using a Graphing Calculator
You can use the STAT program on a graphing calculator to find the line-of-fit for any set of data.
Steps for Calculator
Entering Data –
1. Press the STAT button
2. Select option 1  Edit…
3. Enter the domain under L1 and the range under L2
Finding the Line-of-Fit
1. Press the STAT button
2. Arrow over to CALC
3. Select option 4  LinReg (ax + b)
4. Press ENTER button
Enter the points from the TABLE in the previous example into a graphing program.
What is the Line of Fit? ( y = ax + b)

_____________________
AST: Lesson 2.5 – Correlation
Correlation Coefficient (r) – measures the strength of the linear relationship between two variables
−1 ≤ 𝑟 ≤ 1
-1
Strong
Negative
Correlation
1
0
No Correlation
Strong
Positive
Correlation
Another example
of NO Correlation
Example 1 – A value of r = .87 generally means….
(a) a strong negative relationship
(b) a strong positive relationship
(c) a weak negative relationship
(d) a weak positive relationship
Some STAT programs give r2:
When given r2, you can calculate r
r= ±√𝑟
Using Your Graphing Calculator
Step 1: Turn Diagnostics On
 Click “2ND” 0 (to get to CATALOG)
 Scroll down to DiagnosticsON
 Hit ENTER twice
Step 2:
 Click STAT and choose 1: Edit…
 Enter data under L1 and L2
 Click STAT and arrow right to CALC
 Arrow down to 4: LinReg(ax + b)
You must use the data given to determine whether the correlation coefficient is positive or negative
Example 2 – Find the correlation coefficient. Then tell whether it is positive or negative.
STAT program: y = ax + b
a = -.594
b = 6.821
r2 = .466
Example 3 – Enter the data below into your calculator’s STAT program with Current Age under L1 and
Expected Life Span under L2. (The same data given in Lessons 2.2 & 2.3)
Current
Age
30
35
40
45
50
55
60
65
Expected
Life Span
76.3
76.6
77.0
77.5
78.1
79.0
80.2
81.7
(a) What is the linear equation and correlation coefficient?
(b) Enter the data again, but put AGE under L2 and Life Span under L1.
What is the new linear equation and correlation coefficient?
Important Note!
In most cases, if you switch the independent and dependent variables… (x, y)  (y, x)
 the line of best fit ___________________________________
 BUT the correlation coefficient _________________________________
AST: Lesson 2.4 – Step Functions
Cost of buying “n” DVDs:
C = 16.99n
Table showing the number of DVDs
available for given amounts of money
(example of discrete data)
Dollars
Available
$10
$20
$30
$40
$50
$60
$70
$80
$90
$100
Scatter Plot of Data
Number
of DVDs
0
1
1
2
2
3
4
4
5
5
Step Functions – functions whose graphs look like steps
Greater Integer Functions (aka the rounding-down or floor function) – all real numbers such that f(x) equals
the greatest integer less than or equal to x
𝑓(𝑥) = ⌊𝑥⌋


𝑜𝑟
𝑓(𝑥) = 𝐼𝑁𝑇(𝑥)
Means  f(x) – the output – can ONLY EQUAL a positive or negative whole number (no
decimals/fractions)
Graphing  the line segment will have a closed point on the left, and open on the right
Example 1 – Evaluate the following.
a. ⌊33.4⌋
b.
−
c. 𝐼𝑁𝑇(2𝜋)
Example 2 – Consider 𝑝(𝑥) = ⌊𝑥 + 0.5⌋. Evaluate the following.
a. p(4.3)
b. p(-2.6)
c. p(9)
X
Y
-3
-2.5
Example 3 – Graph 𝑦 = ⌊𝑥 − 1⌋ by creating a table of ordered pairs.
Then, state the domain, range and the values at which the function is
not continuous.
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
Least Integer Function (aka the rounding-up or ceiling function) – all real
numbers such that f(x) equals the smallest integer greater than or equal to x
Example 4 – In 1990, the cost of calling Baghdad, Iraq from anywhere in the
was $2.08 for the first minute and $1.40 for each extra minute or part
thereof.
a. Which of the following gives the cost d in dollars as a function of the time t in minutes?
i. 𝐷 = 1.40 − 2.08⌊𝑡 − 1⌋
ii. 𝐷 = 1.40 + 2.08⌊𝑡 − 1⌋
iii. 𝐷 = 2.08 − 1.40⌈𝑡 − 1⌉
iv. 𝐷 = 2.08 + 1.40⌈𝑡 − 1⌉
b. Using the function chosen above, find the cost of a 7 ½ minute phone call.
Example 5 – State the domain and range of each function graphed.
(a)
(b)
U.S.
AST: Lesson 2.6 – Quadratic Models
Quadratic Models – equations of scatter plots based on quadratic functions; graphs are parabolas
𝑦 = 𝑎𝑥 + 𝑏𝑥 + 𝑐
a>0
Range: y > min
a<0
Range: y < max
Domain: All Real #s (−∞ ≤ 𝑥 ≤ ∞)
You can use the following calculations to help graph and/or analyze the quadratic model:
 Max/Min =
,𝑓
 y-intercept: point on the function where x = 0
𝑓(0) = 𝑎(0 ) + 𝑏(0) + 𝑐
 x-intercepts: points on the function where y = 0
𝑥=
−𝑏 ± √𝑏 − 4𝑎𝑐
2𝑎
𝑓(0) = 𝑐  (0, c)
Example 1 – Consider the function 𝑓(𝑥) = 2𝑥 − 9𝑥 + 3
a. Find the x- and y- intercepts.
b. Find the vertex. Is this a max or min of the quadratic?
c. Sketch a graph of 𝑓(𝑥)
d. State the domain and range of the
function.
A well known quadratic equation is a function from Physics. Newton discovered that the height h of an object
at time t after it has been thrown upward with an initial velocity v0 from an initial height h0 satisfies the
formula
1
ℎ = − 𝑔𝑡 + 𝑣 𝑡 + ℎ
2
Acceleration due to gravity (g) – the value used for gravity depends on the units used in the problem
𝑔 = 9.8 𝑚⁄𝑠𝑒𝑐
or
𝑔 = 32 𝑓𝑡⁄𝑠𝑒𝑐
Example 2 – A projectile is shot from a tower 10 ft high with an upward velocity of 100 ft/sec.
a. Find a quadratic model for the relation between height, h, and time, t, after the object is released.
b. How high is the object after 2 seconds?
c. At what time is the ball 26 feet off the ground?
Example 3: You can use your STAT program to find a quadratic equation for
data. Use the data below and the instructions in the box to answer the
questions below.
X
0.9
1.3
1.3
1.4
1.6
1.8
2.1
2.1
2.5
2.9
Y
2.5
4.03
4.1
4.4
5.1
6.05
7.48
7.6
9.8
12.4
1. Enter Data under (L1, L2)
2. Click STAT and arrow
over to CALC
3. Choose 5: QuadReg
a. Find the quadratic model to represent the relationship between x and y.
b. Predict the output when x = 2. Is this interpolation or extrapolation of the data?