Integrated Math III - Unit 2 Study Guide

Unit 2 – STUDY GUIDE 24-25 Integrated Math III End of Unit 4 Summative Assessment

Standards

  • A.APR.7: Rational expressions form a system analogous to rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.
  • A.CED.1: Create equations and inequalities in one variable and use them to solve problems.
  • A.CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
  • A.REI.2: Solve simple rational and radical equations in one variable and give examples showing how extraneous solutions may arise.
  • A.REI.11: Explain why the x-coordinates of the points where the graphs of the equations y=f(x)y=f(x) and y=g(x)y=g(x) intersect are the solutions of the equation f(x)=g(x)f(x)=g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x)f(x) and/or g(x)g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
  • F.TF.1: Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.
  • F.TF.3: Use special triangles to determine geometrically the values of sine, cosine, tangent for π3\frac{\pi}{3}, π4\frac{\pi}{4}, and π6\frac{\pi}{6}, and use the unit circle to express the values of sine, cosine, and tangent for π−x\pi-x, π+x\pi+x, and 2π−x2\pi-x, in terms of their values for xx, where xx is any real number.
  • F.TF.7: Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

End of Term 2 Final Exam Learning Objectives

  • Simplify rational expressions.
  • Simplify rational expressions by multiplying and dividing.
  • Simplify rational expressions by adding and subtracting.
  • Simplify complex fractions.
  • Recognize and solve direct and joint variation equations.
  • Recognize and solve inverse and combined variation equations.
  • Solve rational equations in one variable.
  • Solve rational inequalities in one variable.
  • Draw angles in standard position and identify coterminal angles.
  • Convert between degree and radian measures and find arc lengths by using central angles.
  • Find values of trigonometric functions for acute angles.
  • Find values of trigonometric functions of general angles.
  • Find values of trigonometric functions by using reference angles.
  • Find values of trigonometric functions given a point on a unit circle or the measure of a special angle.
  • Find values of angle measures by using inverse trigonometric functions.

Resources

  • Required textbook: Reveal Math III by McGraw Hill Education
  • Additional:
    • PowerPoint presentations for Unit 2 (Modules 7 and 9) (Your channel on Teams)
    • Module Tests from McGraw Hill
    • Notebooks and Formatives/Quizzes
    • Online Educational Platform: Khan Academy

Module 7 – Practice Questions

  • 1. Define rational expressions and provide an example.
    • A rational expression is a fraction where both the numerator and the denominator are polynomials.
    • Example: x6−3x2+1\frac{x^6-3}{x^2+1}
  • 2. Identify the four operations that can be performed on rational expressions.
    • Addition, Subtraction, Multiplication, Division
  • 3. List the conditions under which a rational expression is undefined.
    • A rational expression is undefined when the denominator equals zero, since division by zero is undefined.
  • 4. Define extraneous solutions in the context of rational equations.
    • Extraneous solutions are values that solve a transformed equation but do not satisfy the original equation, often because they make the denominator zero.
  • 5. Simplify 3x2+22x+243x2−8x−16\frac{3x^2+22x+24}{3x^2-8x-16}.
    • x+6x−4\frac{x+6}{x-4}
  • 6. Simplify 63a4b245⋅5b69a7\frac{63a^4b^2}{45} \cdot \frac{5b^6}{9a^7}.
    • 7b89a3\frac{7b^8}{9a^3}
  • 7. Estimate the solution of 2x+3=9 and verify it.
    • Estimate: x=3
    • Verify: 2(3)+3=6+3=9
  • 8. Identify whether the expression is defined or undefined for each of the given values of x. x2+5x+6x2−4\frac{x^2 +5x + 6}{x^2 - 4}
    • Defined for x = -3, -1, 1, 3
    • Undefined for x = -2, 2
  • 9. Simplify x2−5xx+6÷x2+3x−40x2−36\frac{x^2-5x}{x+6} \div \frac{x^2+3x-40}{x^2-36}.
    • x(x−6)x+8\frac{x(x-6)}{x+8}
  • 10. Simplify 5x−13x2+9x+8−3x+1\frac{5x-13}{x^2+9x+8} - \frac{3}{x+1}.
    • 2x−37(x+8)(x+1)\frac{2x-37}{(x+8)(x+1)}
  • 11. Simplify a9b+14a\frac{a}{9b} + \frac{1}{4a}.
    • 4a2+9b36ab\frac{4a^2+9b}{36ab}
  • 12. Suppose that y varies directly as x and inversely as z. When x = 8 and z = 10, y = 7. What is the value of y when x = 2 and z = 14?
    • 1.25
  • 13. The table shows the time it took Rachel to type responses to several short answer questions during an online test.
    • Word count: 50, 500, 900
    • Time (minutes): 1.25, 12.5, 22.5
    • Let x be the number of words in the response and let y represent the time, in minutes, required to type the response. The quantity y varies directly as the quantity x. The constant of variation is 0.025.
  • 14. In still water, a boat can travel at 4.5 miles per hour. Suppose a boat travels 8 miles up a river, then returns to its starting location. The complete trip takes 9 hours. What is the average speed of the current, in miles per hour?
    • 3.5 miles per hour

Module 9 – Practice Questions

  • 1. What is the measure of an angle in standard position that is coterminal with 405°?
    • 45°
  • 2. Convert 5π6\frac{5\pi}{6} radians to degrees.
    • 150°
  • 3. If an arc length is 5π\pi units and the radius is 10 units, what is the central angle in radians?
    • π2\frac{\pi}{2}
  • 4. Select a positive and a negative angle measure coterminal with a 65° angle.
    • Ⓐ –295°
    • Ⓕ 425°
  • 5. Which degree measure is equivalent to 4π3\frac{4\pi}{3} radians?
    • 240°
  • 6. The blades of a ceiling fan have a length of 31 inches. The tips of the blades follow a circular path. How far does the tip of a blade travel in 112\frac{1}{12} of a rotation? Round to the nearest hundredth of an inch.
    • 16.23 inches
  • 7. Select the correct trigonometric ratios for angle θ.
    • Ⓐ sin θ = 1010\frac{\sqrt{10}}{10}
    • Ⓒ tan θ = 13\frac{1}{3}
    • Ⓓ csc θ = 10\sqrt{10}
    • Ⓕ cot θ = 3
  • 8. What is the period of this function?
    • 24
  • 9. What are the period and midline of the graph?
    • period: 8
    • midline: y = 2
  • 10. Find sin^{-1}(12\frac{1}{2}).
    • π6\frac{\pi}{6}
  • 11. A carpenter needs to make a diagonal cut in a rectangular board as shown. To the nearest degree, what is the measure of the obtuse angle formed after the triangular piece is cut from the board?
    • 144°