2.1 relations and functions - mrs. scherling's math page · 2018. 9. 5. · name_____ 2.1...

Name_______________________ 2.1 –Relations and Functions

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2.1 – 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. (Remember: Domain is x-values, Range is y-values) Example: State the domain and range of the following relations. Do the relations represent a function?

Domain: {-1, 0, 1, 2, 3} Range: {-5, -3, -1, 1, 3} Each element of the domain corresponds with exactly one element of the range, so it is a function. Domain: {4, 2, 5}. Range: {-3, -1, 1, 0} But, an element of the domain corresponds with 2 elements of the

range. Therefore, this is NOT a function Examples: State the domain and range of each relation. Then determine whether each relation is a function. 1. {(1.5, 2), (3, 2), (-4, 1), (2, -1)} 2. {(-2, 4), (3,1), (2, 0)}

Exercises: State the domain and range of each relation. Then determine whether each relation is a function. 1. {(0.5, 3), (0.4, 2), (3.1, 1), (0.4, 0)} 2. {(-5.2, 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)}

x y

-1 -5

0 -3

1 -1

2 1

3 3

x y

4 -3

2 -1

4 1

5 0


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5. {(4.5, 0), (3, 2), (-3, 1), (4.5, 7)} 6. {(-2, 4), (3, 4), (5, 4), (-1.5, 4)} 7. {(-1, 2), (-14, 5), (3, -1)} 8. {(0, 6), (4.3, -6), (2, 6), (3, -6)} 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 dependant 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.

Examples: given xxxf 2)( 2 , find each value.

1. f(3) 2. f(5a) 3. f(0)


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4. f(-3) 5. f(-5) 6. f(a+1)

The Vertical Line Test If no vertical line intersects a graph in more than one point, the graph represents a function If a vertical line intersects a graph in two or more points, the graph does not represent a function

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. *A relation in which the domain is a set of individual points is said to be a discrete relation. The graph consists of points that are not connected. *When the domain of a relation has an infinite number of elements and the relation can be graphed with a line or smooth curve, the relation is a continuous relation


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Find each value if 42)( xxf .

4. )12(f 5. )6(f 6. )2( bf

Find each value if xxxg 3)( .

7. )5(g 8. )2(g 9. )7( cg

10. The ordered pairs (1, 16), (2, 16), (3, 32) and (5, 48) represent the cost of buying various numbers of CDs through a music club. Identify the domain and range of the relation. Is the relation discrete or continuous? Is the relation a function.


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Name_______________________ 2.2 – Linear Relations and Functions

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2.2 – 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 bmxxf )( , 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 y-intercept and connect these two points with a line Examples:

1. Is 5

2.0)(x

xf a linear function? 2. Is 032 yxyx a linear

Explain. function? Explain. Exercises: State whether each equation or function is a linear function. Write yes or no. Explain.

1. 76 xy 2. y

x18

9 3. 11

2)(x

xf

4. 046

2 x

y 5. 44.26.1 yx 6. y

x4.0

1002.0

7. 34)( xxf 8. x

xf4

)( 9. 0232 xyyx


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The standard form of a linear equation is Ax + By = C, where A, B, and C are integers whose greatest common factor is 1. Also, A must be positive.

Examples: Write each equation in standard form. Identify A, B, and C. 1. 58 xy y 2. 21714 yx 3. yx 35

Example: Find the x-intercept and the y-intercept of the graph 4x – 5y = 20. Then Graph the equation. ***NOTE: The x-intercept is the value of x when y = 0

The y-intercept is the value of y when x = 0

Example: Find the x-intercept and the y-intercept of the graph -3y = -4x + 12. Then Graph the equation


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Exercises: Write each equation in standard form. Identify A, B, and C. 1. 142 yx 2. 325 xy 3. 253 yx

4. 92418 xy 5. 53

2

4

3 xy 6. 01086 xy

7. 1034.0 yx 8. 74 yx 9. 632 xy

Find the x-intercept and the y-intercept of the graph of each equation. Then graph the equation using the intercepts. 10. 1472 yx 11. 105 xy 12. 05.755.2 yx

Name ____________________ 2.3 – Rate of Change and Slope

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2.3 – Rate of Change and Slope 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 y change in x

= change in Elevation of the Sun

Change in Time = _____________ = __________

= degrees per hour _______________________________________________________ Exercises: Find the rate of change for each set of data

Time Elevation of the Sun (in degrees)

7:00 A.M. 6

8:00 A.M. 26

9:00 A.M. 45

10:00 A.M. 64

11:00 A.M. 84


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SLOPE The slope of a line is the ratio of the change in y-coordinates to the corresponding change in the x-coordinates. The slope of a line is the same as its rate of change.

Slope m of a Line

For points ),( 11 yx and ),( 22 yx , where 21 xx ,

12

12

xx

yy

xinchange

yinchangem

Example 1: Find the slope of the Example 2: Find the slope of the line that passes through (2, -1) line. and (-4, 5)

Exercises: 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. (7, -2) and (3, 3)


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Determine the rate of change of each graph (RISE over RUN)

Name _________________________ 2.4 – Writing Linear Equations

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2.4 – Writing Linear Equations

FORMS OF EQUATIONS

Slope-Intercept Form of a Linear Equation

bmxy , where m is the slope and b is the y-intercept

Point-Slope Form of A Linear Equation

)( 11 xxmyy , where ),( 11 yx are the coordinates of

a point on the line and m is the slope of the line

Example 1: Write an equation in slope-intercept form for the line that has slope -2 and passes through the point (3, 7).

Slope intercept form:

Substitute for m, x and y:

Solve for b:

Write the equation:

Example 2: Write an equation in slope-intercept form for the line that has slope

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1 and x-intercept 5.

Write an equation in slope-intercept form for the line described.

1. slope -2, passes through (-4, 6) 2. slope 3

2, y-intercept 4

3. slope 1, passes through (2, 5) 4. m= 5

13 , passes through (5, -7)

Exercises


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Write an equation in slope-intercept form for each graph.

PARALLEL AND PERPENDICULAR LINES

We can use the slope-intercept or point-slope form to find equations of lines that are parallel or perpendicular to a given line. Remember that parallel lines have equal slope. The slopes of two perpendicular lines are negative reciprocals, that is, their product is -1. Example 1: Write an equation of the Example 2: write an equation of the line line that passes through (8, 2) and that passes through (-1, 5) and is Is perpendicular to the line those parallel to the graph of 13 xy .

equation is 32

1 xy .


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Write an equation in slope-intercept form for the line that satisfies each set of conditions

1. passes through (-4, 2), parallel to 52

1 xy

2. passes through (3, 1), perpendicular to 23 xy

3. passes through (1, -1), parallel to the line that passes through (4, 1) and (2,-3)

4. passes through (4, 7), perp. to the line that passes through (3, 6) and (3, 15)

5. passes through (8, -6), perpendicular to 42 yx

6. passes through (2, -2), perpendicular to 65 yx

7. passes through (6, 1), parallel to the line with x-int. -3 and y-int 5

8. passes through (-2, 1), perpendicular to 114 xy

Exercises

Name _________________________ 2.6 – Special Functions

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2.6 – Special Functions PIECEWISE-DEFINED FUNCTIONS A piecewise-defined function is written using two or more expressions. Its graph is often disjointed.

Example: Graph

21

22)(

xifx

xifxxf

First, graph the linear function f(x) = 2x for x < 2. Since 2 does not satisfy this inequality, stop with a circle at (2, 4). Next, graph the linear function f(x) = x – 1 for x ≥ 2. Since 2 does satisfy this inequality, begin with a dot at (2, 1).

Graph each function. Identify the domain and range.

1.

2.

Exercises


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STEP FUNCTIONS AND ABSOLUTE VALUE FUNCTIONS

Name Written as Graphed as

Greatest Integer Function xxf )(

Absolute Value Function xxf )( two rays that are mirror images of each other and meet at a point, the vertex

Example: Graph 43)( xxf

Find several ordered pairs. Graph the points and connect them. You would expect the graph to look similar to its

parent function, xxf )(

Example: Graph 2)( xxf

Greatest Integer Function: rounds down a real number to the nearest integer.

x 43 x

0

1

2

-1

-2

x 2x

0

0.5

1

-0.5

-1


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HOW TO GRAPH ABSOLUTE VALUE EQUATIONS by finding the vertex 1) Set the inside of the absolute value equal to 0 and solve for x. 2) Substitute the value of x into the original equation and solve for y. 3) The ordered pair (x, y) is the vertex of the absolute value 4) Pick x-values that are close to the vertex and plug them in and solve for y.

Examples:

1. 25 xy 2. 232 xy

HW: Pg. 104, #1-4, 6-11 (Take graph paper to graph some of the problems!)

Name _________________________ 2.8 – Graphing Linear and Absolute Value Inequalities

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2.8 – Graphing Linear and Absolute Value Inequalities

A linear inequality is an inequality whose graph is a region of the coordinate plane that is bounded by a line. A solution of a linear inequality in x and y is an ordered pair (x, y) if the linear inequality is TRUE when the values of x and y are substituted into the inequality. Example: 2x + 5y > 6 a) Is (1, 2) a solution? b) Is (4, -3) a solution? Graphing a Linear Inequality:

1) Sketch the line given by the corresponding linear equation

2) Use a SOLID LINE for

Use a DOTTED LINE for < or > Examples:

3y 824 yx


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42 xy 42 yx

Graphing Absolute Value Inequalities is similar to graphing linear inequalities.

EXAMPLE: Graph 13 xy

First graph the equation 13 xy

Since the inequality is ≤, the graph of the boundary is solid. Test (0, 0)

Shade the region that contains (0, 0)

Graph each inequality and list the vertex.

EXERCISES


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2.1 relations and functions - mrs. scherling's math page · 2018. 9. 5. · name_____ 2.1...

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