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Graphs of Functions Defined by Expressions in a Linear Equation
On a standard screen, graph the following functions, determined from the given linear equation:
1. 2.
Describe the characteristic of the graph of the function defined by the expression in Y1. Describe the characteristic of the graph of the function defined by theexpression in Y2.
3 8 6
Y1 3 8 Y2 6
x
x
2 5 3 4
Y1 2 5 Y2 3 4
x x
x x
1 of 2
Chapter 4Discovery 1
Graphs of Functions Defined by Expressions in a Linear Equation
Describe the characteristic of the graph of the function defined by the expression in Y1.
Describe the characteristic of the graph of the function defined by the expression in Y2.
3
3
2
2Y1 Y2
x xx
x xx
On an integer screen, graph the following functions, determine from the given nonlinear equation:
3.
2 of 2
Chapter 4Discovery 1
Numerical Solutions
To solve the equation, , complete the extended table of values shown, compare the values obtained, and determine the difference of the values.
x =
0
1
2
3
4
6
Write a rule to determine the solution of an equation from a table of values.
3 9 2 6 x x
3 9x 2 6 x9 6 9 6 15 9
Chapter 4Discovery 2
Linear Equations with Noninteger Solutions
(5 4) 2(3 1) 2( 7) x x x does not have an integer solution. Complete the table of values, compare the values obtained, and determine their differences.
x (5 4) 2(3 1) 2( 7) x x x
1 83
4
5
6
7
1 8 1 ( 8) 7
Write a rule for determining when the solution of an equation is between two integers given in a table of values.
Chapter 4Discovery 3
Graphical Solutions
To solve the equation , graph the functions and . Label the point of intersection of the graphs.
3 9 2 6 x x
Write a rule for determining the solution of an equation from the graph of the two functions.
Write a rule for determining the numeric value of each expressionwhen the equation is evaluated at its solution.
Y2 2 6 xY1 3 9 x
Chapter 4Discovery 4
Linear Equations with No Solution
1. Solve numerically by completing a table of values.
Write a rule explaining how to solve the equation by viewing its table.
2. Solve graphically. Sketch the graph.
Write a rule explaining how to solve the equation by viewing its graph.
2 5 2 10 x x
2 5 2 10 x x
Chapter 4Discovery 5
Linear Equation with Many Solutions
1. Solve numerically by completing a table of values.
Write a rule explaining how to solve the equation by viewing its table of values.
2. Solve graphically. Sketch the graph.
Write a rule explaining how to solve the equation by viewing its graph.
2 5 ( 3) ( 2) x x x
Chapter 4Discovery 6
2 5 ( 3) ( 2) x x x
Addition Property of Equations
Given the equation 7 = 7, add 2 to both expressions.
7 = 7
7 + 2 7 + 2
9 91. Given the equation 7 = 7, add -2 to both expressions.
2. Given the equation 6 + 1 = 4 + 3, add 2 to both expressions.
3. Given the equation 6 + 1 = 4 + 3, add -2 to both expressions.
Write a rule for the addition property of equations.
Chapter 4Discovery 7
Multiplication Property of Equations
Given the equation 7 = 7, multiply both expressions by 2.
7 = 77 • 2 7 • 2
14 14
1. Given the equation 7 = 7, multiply both expressions by -2.
2. Given the equation 6 + 1 = 4 + 3, multiply both expressions by 2.
3. Given the equation 6 + 1 = 4 + 3, multiply both expressions by -2.
Write a rule for the multiplication property of equations.
Chapter 4Discovery 8
Linear Equation with No Solution
Solve algebraically the previous example of a linear equation with no solution:
2 5 2 10 x x
Write a rule that explains why the equation has no solution.
Chapter 4Discovery 9
Linear Equations with Many Solutions
Solve algebraically the previous example of a linear equation with many solutions:
Write a rule that explains why the equation has many solutions.
2 5 ( 3) ( 2) x x x
Chapter 4Discovery 10
Graphs of the Function Defined by Expressions in a Linear Absolute-Value Equation
On an integer screen, graph the following functions, determined from the given linear absolute-value equation:
2 3 5
Y1 2 3 Y2 5
x
x
1. 2. 2 3 5
Y1 2 3 Y2 5
x
x
Explain the characteristic of the graphs of the functions defined by the absolute-value expression, Y1.
Explain the characteristic of the graphs of the functions defined by the constant, Y2.
1 of 2
Chapter 4Discovery 11
Graphs of the Function Defined by Expressions in a Linear Absolute-Value Equation
2 of 2
On an integer screen, graph the following functions, determined from the given linear absolute-value equation:
. 2 3 6 5
Y1 2 3 6 Y2 5
3 x
x
. 4 2 3 0
Y1 4 2 3 Y2 0
4 x
x
Explain the characteristic of the graphs of the functions defined bythe absolute-value expression, Y1.
Explain the characteristic of the graphs of the functions defined bythe constant, Y2.
Chapter 4Discovery 11
Solving a Linear Absolute-Value Equation
Solve each equation graphically and check your solution numerically.
1. a. b.
4 3
3 2
x
x
2. a. b.
4 3
3 2
x
x
3. a. b.
4 0
3 0
x
x
Write a rule for the number of solutions of a linear absolute-value equation when the absolute-value expression equals a positive number, a negative number, and 0.
Chapter 4Discovery 12
