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Contents Lesson 12-1Linear and Nonlinear Functions Lesson 12-2Graphing Quadratic Functions Lesson 12-3Simplifying Polynomials Lesson 12-4Adding Polynomials Lesson 12-5Subtracting Polynomials Lesson 12-6Multiplying and Dividing Monomials Lesson 12-7Multiplying Monomials and Polynomials

Lesson 1 Contents Example 1Identify Functions Using Graphs Example 2Identify Functions Using Graphs Example 3Identify Functions Using Equations Example 4Identify Functions Using Equations Example 5Identify Functions Using Tables Example 6Identify Functions Using Tables Example 7Identify Functions Using Tables

Example 1-1a Determine whether the graph represents a linear or nonlinear function. Explain. Answer: The graph is a curve, not a straight line. So it represents a nonlinear function.

Example 1-1b Determine whether the graph represents a linear or nonlinear function. Explain. Answer: Nonlinear; the graph is a curve.

Example 1-2a Determine whether the graph represents a linear or nonlinear function. Explain. Answer: The graph is a straight line. So it represents a linear function.

Example 1-2b Determine whether the graph represents a linear or nonlinear function. Explain. Answer: Linear; the graph is a straight line.

Example 1-3a Determine whether represents a linear or nonlinear function. Explain. Answer: Since x is raised to the second power, the equation cannot be written in the form

Example 1-3b Determine whether represents a linear or nonlinear function. Explain. Answer: Nonlinear; the power of x is greater than one.

Example 1-4a Determine whether represents a linear or nonlinear function. Explain. Answer: This equation is linear since it is of the form

Example 1-4b Determine whether represents a linear or nonlinear function. Explain. Answer:

Example 1-5a Determine whether the table represents a linear or nonlinear function. Explain. As x increases by 2, y increases by a greater amount each time. Answer: The rate of change is not constant, so this function is nonlinear. x y

Example 1-5b Determine whether the table represents a linear or nonlinear function. Explain. x1357 y Answer: Linear; the rate of change is constant, as x increases by 2, y increases by 4.

Example 1-6a Determine whether the table represents a linear or nonlinear function. Explain. x14710 y As x increases by 3, y increases by 9 each time. Answer: The rate of change is constant, so this function is linear

Example 1-6b Determine whether the table represents a linear or nonlinear function. Explain. x1234 y Answer: Nonlinear; the rate of change is not constant.

Example 1-7a CLOCKS Use the table below to determine whether or not the number of revolutions per hour that the minute hand on a clock makes is a linear function of the number of hours that pass. HourMinute Hand Revolutions Examine the difference between the minute hand revolutions for each hour. Answer: The differences are the same, so the function is linear.

Example 1-7b GEOMETRY Use the table below to determine whether or not the sum of the measures of the angles in a polygon is a linear function of the number of sides. Number of Sides Sum of the Angles      Answer: linear

End of Lesson 1

Lesson 2 Contents Example 1Graph Quadratic Functions: y = ax 2 Example 2Graph Quadratic Functions: y = ax 2 Example 3Graph Quadratic Functions: y = ax 2 + c Example 4Graph Quadratic Functions: y = ax 2 + c Example 5Graph a Function to Solve a Problem

Example 2-1a Graph To graph a linear function, make a table of values, plot the ordered pairs, and connect the points with a smooth curve. (2, 20)20 2 (1, 5) 5 1 (0, 0) 0 0 (–1, 5) 5–1 (–2, 20)20–2 (x, y)y5x25x2 x

Example 2-1a Answer: y = 5x 2

Example 2-1b Graph Answer:

Example 2-2a (2, –16)–16 2 (1, –4) –4 1 (0, 0) 0 0 (–1, –4) –4–1 (–2, –16)–16–2 (x, y)y–4x 2 x Graph

Example 2-2a Answer: y = –4x 2

Example 2-2b Graph Answer:

Example 2-3a (2, 13)13 2 (1, 4) 4 1 (0, 1) 1 0 (–1, 4) 4–1 (–2, 13)13–2 (x, y)y3x 2 + 1x

Example 2-3a Answer: y = 3x 2 + 1

Example 2-3b Answer:

Example 2-4a (2, –6)–6 2 (1, –3)–3 1 (0, –2)–2 0 (–1, –3)–3–1 (–2, –6)–6–2 (x, y)y–x 2 – 2x

Example 2-4a Answer: y = –x 2 – 2

Example 2-4b Answer:

Example 2-5a GRAVITY The function describes the distance d in meters that a rock falls from a high cliff during time t. Graph this function. Then use your graph to estimate how long it would take a rock to fall 400 meters. The equation is quadratic, since the variable t has an exponent of 2. Time cannot be negative, so use only positive values of t. (4, 78.4)4 (3, 44.1)3 (2, 19.6)2 (1, 4.9)1 (0, 0)0 (t, h)h = 4.9t 2 t

Example 2-5a Answer: The rock will have fallen 400 meters in about 9 seconds.

GRAVITY The function describes the height h in meters that a rock is during a fall from a high building at time t. Graph this function. Then use your graph to estimate how long it would take a rock to be at a height of 77 meters. Example 2-5b Answer: It would take a rock approximately 5 seconds to fall from a height of 77 meters.

End of Lesson 2

Lesson 3 Contents Example 1Simplify a Polynomial Example 2Simplify Polynomials Example 3Simplify Polynomials

Example 3-1a The like terms in this expression are 3r and –r. Write the polynomial. Definition of subtraction Group like terms. Simplify by combining like terms. Answer:

Example 3-1b Answer:

Example 3-2a There are no like terms in the expression. Answer:

Example 3-2b Answer: simplest form

Example 3-3a Method 1 Use models.

Example 3-3a Group tiles with the same shape and remove zero pairs.

Example 3-3a Method 2 Use symbols. Write the polynomial. Then group and add like terms. Answer:

Example 3-3b Answer:

End of Lesson 3

Lesson 4 Contents Example 1Add Polynomials Example 2Add Polynomials Example 3Add Polynomials Example 4Add Polynomials Example 5Use Polynomials to Solve a Problem

Example 4-1a Method 1 Add vertically. Method 2 Add horizontally. Answer: The sum is Align like terms. Add. Associative and Commutative Properties

Example 4-1b Answer:

Example 4-2a Method 1 Add vertically. Method 2 Add horizontally. Answer: The sum is

Example 4-2b Answer:

Example 4-3a Answer: The sum is Group like terms. Simplify.

Example 4-3b Answer:

Example 4-4a Leave a space because there is no other term like 3x 2. Answer: The sum is

Example 4-4b Answer:

Example 4-5a Read the Test Item The figure is a triangle. The sum of the measures of the angles of a triangle equals 180 . The measure of each angle is determined by x. MULTIPLE-CHOICE TEST ITEM Find the measure of  A in the figure below. A 13  B 38  C 53  D 109 

Example 4-5a Solve the Test Item Write an equation to find the value of x. The sum of the measures of the angles equals180 Write the equation. Group like terms. Simplify. Add 10 to each side. Simplify. Divide each side by 5. Simplify.

Example 4-5a Find the measure of  A. Write the expression for the measure of angle A. Replace x with 38. Simplify. The measure of  A is 53 . Answer: C

Example 4-5b MULTIPLE-CHOICE TEST ITEM Find the measure of  A in the figure below. A 40  B 50  C 90  D 11  Answer: B

End of Lesson 4

Lesson 5 Contents Example 1Subtract Polynomials Example 2Subtract Polynomials Example 3Subtract Using the Additive Inverse Example 4Subtract Using the Additive Inverse Example 5Use Polynomials to Solve a Problem

Example 5-1a Align the terms. Subtract. Answer: The difference is

Example 5-1b Answer:

Example 5-2a Align the terms. Subtract. Answer: The difference is

Example 5-2b Answer:

Example 5-3a Group like terms. Simplify by combining like terms. The additive inverse of Answer: The difference is

Example 5-3b Answer:

Example 5-4a Answer: The difference is The additive inverse of

Example 5-4b Answer:

Words Example 5-5a Write an expression for the difference of the distances traveled by each marble. EXPERIMENTS Students are rolling identical marbles down two side-by-side ramps. The marble on ramp A rolls inches in t seconds. The marble on ramp B rolls inches in t seconds. How far apart are the marbles after 6 seconds? Variables Marble A’s distance minus marble B’s distance Expression

Example 5-5a Now evaluate this expression for a time of 6 seconds. Replace t with 6. Simplify. Answer: After 6 seconds, the cars are 78 inches apart.

Example 5-5b Answer: 125 in. EXPERIMENTS Students are rolling identical marbles down two side-by-side ramps. The marble on ramp A rolls inches in t seconds. The marble on ramp B rolls inches in t seconds. How far apart are the marbles after 5 seconds?

End of Lesson 5

Lesson 6 Contents Example 1Multiply Powers Example 2Multiply Monomials Example 3Divide Powers Example 4Divide Powers Example 5Divide Powers to Solve a Problem

Example 6-1a The common base is 7. Add the exponents. Check Answer: Find. Express using exponents.

Example 6-1b Answer:

Find. Express using exponents. Example 6-2a Answer: Commutative and Associative Properties The common base is x. Add the exponents.

Example 6-2b Find. Express using exponents. Answer:

Example 6-3a The common base is 6. Simplify. Answer: Find. Express using exponents.

Example 6-3b Answer: Find. Express using exponents.

Example 6-4a The common base is a. Simplify. Answer: Find. Express using exponents.

Example 6-4b Answer: Find. Express using exponents.

Example 6-5a UNIT CONVERSION One centimeter is 10 millimeters, and one kilometer is 10 6 millimeters. How many centimeters are there in one kilometer? To find how many centimeters there are in one kilometer, divide 10 6 by 10. Quotient of Powers Simplify. Answer: There are 10 5 centimeters in one kilometer.

Example 6-5b Answer: 10 4 decimeters UNIT CONVERSION One decimeter is 10 centimeters, and one kilometer is 10 5 centimeters. How many decimeters are there in one kilometer?

End of Lesson 6

Lesson 7 Contents Example 1Use the Distributive Property Example 2Use the Distributive Property Example 3Use the Product of Powers Rule Example 4Use the Product of Powers Rule

Example 7-1a Distributive Property Answer:

Example 7-1b Answer:

Example 7-2a Distributive Property Definition of subtraction Answer:

Example 7-2b Answer:

Example 7-3a Distributive Property Answer:

Example 7-3b Answer:

Example 7-4a Distributive Property Simplify. Definition of subtraction Answer:

Example 7-4b Answer:

End of Lesson 7

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