Name: __________________________________________Date: __________________________________________ Period: ______
Unit 2: Linear Relations and Functions
NAME _____________________________________________ DATE ____________________________ PERIOD _____________
2-1a Study Guide and Intervention
Relations and Functions
Relations and Functions A relation can be represented as a set of ordered pairs or as an equation; the relation is then
the set of all ordered pairs (x, y) that make the equation true. A function is a relation in which each element of the domain
is paired with exactly one element of the range.
One-to-One Each element of the domain pairs to exactly one unique element of the range.
Function
Onto Function Each element of the range also corresponds to an element of the domain.
Both One-to- Each element of the domain is paired to exactly one element of the range and
One and Onto each element of the range.
Example: State the domain and range of the relation. xy
Does the relation represent a function? –1 –5
0 –3
The domain and range are both all real numbers. Each element of the domain 1 –1
corresponds with exactly one element of the range, so it is a function. 21
33
Exercises
State the domain and range of each relation. Then determine whether each relation is a function. If it is a function,
determine if it is one-to-one, onto, both, or neither.
1. {(0.5, 3), (0.4, 2), (3.1, 1), (0.4, 0)} 2. {(–5, 2), (4, –2), (3, –11), (–7, 2)}
3. {(0.5, –3), (0.1, 12), (6, 8)} 4. {(–15, 12), (–14, 11), (–13, 10), (–12, 12)}
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Name: __________________________________________Date: __________________________________________ Period: ______
2-1a Skills Practice (continued)
Relations and Functions
State the domain and range of each relation. Then determine whether each relation is a function. If it is a function,
determine if it is one-to-one, onto, both or neither.
1. 2.
3. x y 4.
12
24
36
Graph each relation or equation and determine the domain and range. Determine whether the equation is a
function, is one-to-one, onto, both, or neither. Then state whether it is discrete or continuous.
5. {(2, –3), (2, 4), (2, –1)} 6. {(2, 6), (6, 2)}
7. {(–3, 4), (–2, 4), (–1, –1), (3, –1)} 8. x = –2
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NAME _____________________________________________ DATE ____________________________ PERIOD _____________
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2-1a Word Problem
Practice
Relations and Functions 3. SCHOOL The number of students N in Vassia’s school
is given by N = 120 + 30G, where G is the grade level. Is
1. PLANETS The table below gives the mean distance 285 in the range of this function?
from the Sun and orbital period of the eight major planets
in our Solar System. Think of the mean distance as the
domain and the orbital period as the range of a relation. Is
this relation a function? Explain. 4. FLOWERS Anthony decides to decorate a ballroom
Planet Mean Distance from Orbital Period with r = 3n + 20 roses, where n is the number of dancers.
Mercury Sun (AU) (years) It occurs to Anthony that the dancers always come in
pairs. That is, n = 2p, where p is the number of pairs.
0.387 0.241 What is r as a function of p?
Venus 0.723 0.615
Earth 1.0 1.0 5. SALES Cool Athletics introduced the new Power
Mars 1.524 1.881 Sneaker in one of their stores. The table shows the sales
Jupiter 5.204 11.75 for the first 6 weeks.
Saturn 9.582 29.5 Week 12 3 4 5 6
Uranus 19.201 84
Neptune 30.047 165 Pairs Sold 8 10 15 22 31 44
a. Graph the data.
2. PROBABILITY Martha rolls a number cube several
times and makes the frequency graph shown. Write a
relation to represent this data.
b. Identify the domain and range.
c. Is the relation a function? Explain.
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Name: __________________________________________Date: __________________________________________ Period: ______
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2-1b Study Guide and Intervention
Evaluating Functions
Equations of Relations and Functions Equations that represent functions are often written in functional notation. For
example, y = 10 – 8x can be written as f(x) = 10 – 8x. This notation emphasizes the fact that the values of y, the dependent
variable, depend on the values of x, the independent variable.
To evaluate a function, or find a functional value, means to substitute a given value in the domain into the equation to find
the corresponding element in the range.
Example: Given f(x) = + 2x, find each value.
a. f(-3) b. f( − )
f(x) = 2 + 2x Original function f(x) = 2 + 2x Original function
f(3) = (−3)2 + 2(−3) Simplify. f(5a) = (5 − 1)2 + 2(5 − 1) Substitute.
=9–6=3 = 252 − 10 + 1 + 10 − 2 Simplify.
= 252 − 1
Exercises
Graph each relation or equation and determine the domain and range. Determine whether the relation is a
function, is one–to–one, onto, both, or neither. Then state whether it is discrete or continuous.
1. y = 3 2. y = 2 – 1 3. y = 3x + 2
Find each value if f(x) = –2x + 4.
4. f (12) 5. f (−6) 6. f (2b) 7. f (−3 − 7)
Find each value if g(x) = x3 – x.
8. g (5) 9. g (–2) 10. g (4c) 11. g (−6)
Find each value if f(x) = 2x – 1 and g(x) = 2 – .
10. f (0) 11. f (12) 12. g (52) 13. g (2 − 3)
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NAME _____________________________________________ DATE ____________________________ PERIOD _____________
2-2 Study Guide and Intervention
Linear Relations and Functions
Linear Relations and Functions A linear equation has no operations other than addition, subtraction, and
multiplication of a variable by a constant. The variables may not be multiplied together or appear in a denominator. A
linear equation does not contain variables with exponents other than 1. The graph of a linear equation is always a line.
A linear function is a function with ordered pairs that satisfy a linear equation. Any linear function can be written in the
form f(x) = mx + b, where m and b are real numbers.
If an equation is linear, you need only two points that satisfy the equation in order to graph the equation. One way is to
find the x-intercept and the y-intercept and connect these two points with a line.
Example 1: Is f(x) = 0.2 – a linear Example 2: Is 2x + xy – 3y = 0 a Example 3: Is 2√ + a linear
linear function? Explain.
function? Explain.
function? Explain.
Yes; it is a linear function because it No; it is not a linear function because No; it is not a linear function because
the variables x is inside of a non-linear
can be written in the form f(x) = – 51x + the variables x and y are multiplied function and it is also appearing in the
0.2. together in the middle term. denominator.
Exercises
State whether each function is a linear function. Write yes or no. Explain.
1. 6y – x = 7 2. 9x = 18 3. f (x) = 2 –
11
4. 2y – – 4 = 0 5. 1.6x – 2.4y = 4 6. 0.2x = 100 – 0.4
6
7. f(x) = 4 – 3 8. f(x) = 4 9. 2yx – 3y + 2x = 0
10. 7y + 42 = –8
11. − = 6 12. = 1
2 3
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Name: __________________________________________Date: __________________________________________ Period: ______
2-2 Study Guide and Intervention (continued)
Standard Form of a Linear Equation
Standard Form The standard form of a linear equation is Ax + By = C, where A, B, and C are integers whose greatest
common factor is 1.
Example 1: Write each equation in standard form. Identify A, B, and C.
a. y = 8x – 5 b. 14x = –7y + 21
14x = –7y + 21
y = 8x – 5 Original equation Original equation
–8x + y = –5 Subtract 8x from each side. 14x + 7y = 21 Add 7y to each side.
8x – y = 5 Multiply each side by –1. 2x + y = 3 Divide each side by 7.
So A = 8, B = –1, and C = 5. So A = 2, B = 1, and C = 3.
Example 2: Find the x-intercept and the y-intercept of the graph of 4x – 5y = 20. Then graph the equation. The x-
intercept is the value of x when y = 0.
4x – 5y = 20 Original equation
4x – 5(0) = 20 Substitute 0 for y.
x=5 Simplify.
So the x-intercept is 5. Similarly, the y-intercept is –4.
Exercises
Write each equation in standard form. Identify A, B, and C.
1. 2x = 4y –1 2. 5y = 2x + 3 3. 3x = –5y + 2
4. 18y = 24x – 9 5. 43y = 32x + 5 6. 6y – 8x + 10 = 0
7. 0.4x + 3y = 10 8. x = 4y – 7 9. 2y = 3x + 6
Find the x-intercept and the y-intercept of the graph of each equation. Then graph the equation using the
intercepts.
10. 2x + 7y = 14 11. 5y – x = 10 12. 2.5x – 5y + 7.5 = 0
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NAME _____________________________________________ DATE ____________________________ PERIOD _____________
2-3 Study Guide and Intervention
Rate of Change and Slope
Rate of Change Rate of change is a ratio that compares how much one quantity changes, on average, relative to the
change in another quantity.
Example: Find the average rate of change for the data in the table.
Average Rate of Change = change in Elevation of the Time
change in Sun (in degrees)
7:00 A.M.
= change in Elevation of the Sun 6° 8:00 A.M.
change in Time 26° 9:00 A.M.
45° 10:00 A.M.
= 11:00 84° – 6° A.M. 64° 11:00 A.M.
A.M. – 7:00 84°
= 4 78°
hours
= 19.5 degrees per hour
Exercises
Find the rate of change for each set of data.
1. Time People in 2. Time Altitude of
P.M. Auditorium (minutes) balloon (meters)
7:15 26 3 520
7:22 61 8 1,220
7:24 71 11 1,640
7:30 101 15 2,200
7:40 151 23 3,320
3. Time Vehicles through 4. Time Depth of sinking
(minutes) Tunnel (seconds) stone (meters)
0 3.51
4 1,610 7 4.77
11 5.49
11 2,131 21 7.29
29 8.73
19 2,746
22 2,970
28 3,432
5. Time Water through 6. Time Distance between
(seconds) Channel (liters) (seconds) Two Sleds (meters)
6 22,172 0 37.3
13 24,706 3 30.2
15 25,430 4 27.7
23 28,326 7 20.8
47 37,014 13 7.2
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NAME _____________________________________________ DATE ____________________________ PERIOD _____________
2-3 Study Guide and Intervention (continued)
Rate of Change and Slope
Slope m of a Line For points (1, 1) and (2, 2), where 1 ≠ 2, m = change in = 2− 1
change in 2 − 1
Example 1: Find the slope of the line that passes Example 2: Find the slope of the line.
through (2, –1) and (– 4, 5).
Find two points on the
m = 2− 1 Slope formula line with integer
2 − 1
coordinates, such as
= 5 − (−1) (1, 1) = (2, –1), (2, 2) = (– 4, 5) (1, –2) and (3, –3).
−4 − 2
Divide the difference in
= 6 = –1 Simplify. the y-coordinates by the
−6
difference in the x-
The slope of the line is –1. coordinates:
−3 − (−2) = – 1
3−1 2
1
The slope of the line is – 2
Find the slope of the line that passes through each pair
of points. Express as a fraction in simplest form.
1. (4, 7) and (6, 13) 2. (6, 4) and (3, 4) 3. (5, 1) and (7, –3)
4. (5, –3) and (– 4, 3) 5. (5, 10) and (–1,–2) 6. (–1, – 4) and (–13, 2)
7. (7, –2) and (3, 3) 8. (–5, 9) and (5, 5) 9. (4, –2) and (– 4, –8)
Determine the rate of change of each graph.
10. 11. 12.
15.
13. 14.
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