Algebra 1 Fall Final Exam Study Guide PDF

# Algebra 1 Fall Final Exam Study Guide ## Use the graph for questions 1-7 - **Function or Relation?:** Function - **Continuous or Discrete?:** Continuous ## 2. What is f(1)? -5 ## 3. For which value of x does f(x) = -2? 0 ## 4. Describe the End Behavior - **as x → +∞, y → -∞** - **as x → -...

# Algebra 1 Fall Final Exam Study Guide ## Use the graph for questions 1-7 - **Function or Relation?:** Function - **Continuous or Discrete?:** Continuous ## 2. What is f(1)? -5 ## 3. For which value of x does f(x) = -2? 0 ## 4. Describe the End Behavior - **as x → +∞, y → -∞** - **as x → -∞, y → +∞** ## 5. What is the y-intercept? (0, -2) ## 6. Write the domain in set notation and interval notation? - **Set Notation:** {x | x ∈ R} - **Interval Notation:** (-∞, ∞) ## 7. What is f(-2)? 4 ## 8. If f(x) = x + 2, find f(-4). - f(-4) = (-4) + 2 - f(-4) = -1 + 2 - f(-4) = 1 ## 9. If f(x) = -x + 21, find f(-3) - f(-3) = -(-3) + 21 - f(-3) = 3 + 21 - f(-3) = 24 ## Consider the following sequence: 4, 8, 12, 16 - **a₁ = 4** - **d = 4** ## 10. Write the recursive formula that represents the sequence. - **aₙ = aₙ₋₁ + 4** ## 11. Write the explicit formula that represents the sequence. - **aₙ = 4 + 4(n - 1)** ## 12. Find the 20th term in the sequence. - a₂₀ = 4 + 4(20 - 1) - a₂₀ = 4 + 4(19) - a₂₀ = 4 + 76 - a₂₀ = 80 ## 13. To qualify for a loan from a bank, the total in someone's checking and savings accounts together must be at least $3,400. Write an inequality that represents the situation. - x + y ≥ 3,400 ## Use the graph to answer questions 14-17. ## 14. Choose the inequality whose solution set is represented by the graph. - **x - 3y ≤ 5** ## 15. Is the point (2, -4) a solution to the inequality? How do you know? - No. The point does not lie in the shaded region. ## 16. Is the point (2, -1) a solution to the inequality? How do you know? - No. The point is located on the dotted line. The points on a dotted line are not included in the solution set. ## 17. List two points that are solutions to the inequality. List two points that are not solutions. - **Solutions:** (0, 0), (-2, 5) - **Not Solutions:** (4, -5), (-2, 6) ## 18. Which graph represents the system of inequalities? - **3x + 6y > 18** - **2x - 2y ≥ 12** ## Use the following scenario to answer questions #19-21. - Jonathan has p pennies and n nickels that add up to more than 40 cents. He has no more than 20 coins altogether. The graph below represents his situation. ## 19. - **0.01p + 0.05n > 0.40 ** - **p + n ≤ 20** ## 20. If we know Jonathan has 8 pennies, what is the maximum number of nickels he could have? - 12 nickels ## 21. Which coin combination is possible for Jonathan? - 10 pennies and 8 nickels ## 22. Match the following graphs with the correct inequality statements. - **A:** x > -4 - **B:** y > -4 - **C:** y ≤ -4 - **D:** y ≥ -4 - **E:** x ≤ -4 ## 23. Simplify √169 - 13 ## 24. Simplify ∛48 - ∛(8 * 6) - ∛8 * ∛6 - 2∛6 ## 25. Simplify ∛-27 - ∛(-1 * 27) - ∛-1 * ∛27 - -3 ## 26. Simplify √75 - √(25 * 3) - √25 * √3 - 5√3 ## 27. Simplify ⁴√32 - ⁴√(16 * 2) - ⁴√16 * ⁴√2 - 2⁴√2 ## 28. Simplify ³√24 - ³√(8 * 3) - ³√8 * ³√3 - 2³√3 ## 29. Simplify the following expression: √72 - 2√9 - √(36 * 2) - 2√9 - 6√2 - 6 - 6√2 - 6 = 6√2 - 6 ## 30. Simplify the following expression: -2√28 + √28 - √7 - -2√(4*7) +√(4*7) -√7 - -4√7 + 2√7 - √7 - -3√7 ## 31. What is the sum of 7√45 + 3√20? - 7√(9*5 ) + 3√(4*5) - 21√5 + 6√5 - 27√5 ## 32. Multiply √6(√7+√16) - √6 * √7 + √6 * √16 - √42 + √96 - √42 + √(16 * 6) - √42 + 4√6 ## 33. Mulitply (5√10) (-2√8) - 5 * -2 * √10 * √8 - -10 * √(10 * 8) - -10 * √80 - -10 * √(16 * 5) - -10 * 4√5 - -40√5 ## 34. Find the area of the rectanlge. Simplify your answer. - A = l * w - A = √27 * 2√3 - A = √(9 * 3) * 2√3 - A = 3√3 * 2√3 - A = 6 * 3 - A = 18 ## 35. Which of the following polynomials is a quadratic monomial? - -5x² ## 36. Which of the following polynomials is a linear binomial? - -10x + 1 ## 37. Add (4p² - ⅓p + ⅘) + (p² - ⅔p + ½) - (4p² + p²) + (-⅓p - ⅔p) + (⅘ + ½) - 5p² - p + 13/10 ## 38. Subtract (4p² - 3p + 1) - (p² - 5p + 9) - 4p² - 3p + 1 - p² + 5p - 9 - (4p² - p²) + (-3p + 5p) + (1 - 9) - 3p² + 2p - 8 ## 39. Multiply (x + 3) (2x - 5) - 2x² -5x + 6x - 15 - 2x² + x - 15 ## 40. What is the area of a square with side lengths of 4a - 5? - A = l * w - A = (4a-5) * (4a-5) - A = 16a² - 20a - 20a + 25 - A = 16a² - 40a + 25 ## 41. Factor: x² + 4x + 4 - (x² + 2x) + (2x + 4) - x(x + 2) + 2(x + 2) - (x + 2)(x + 2) - (x + 2)² ## 42. Factor: 3x² - 9x - 12 - **GCF:** 3 - 3(x² - 3x - 4) - **a*c:** -4 * 1 = -4 - 3(x² - 4x + x - 4) - 3[x(x - 4) + 1(x - 4)] - 3(x + 1)(x - 4) ## 43. Factor: x² + 2x - 63 - **a*c:** 1 * -63 = -63 - (x² + 9x) + (-7x - 63) - x(x + 9) - 7(x + 9) - (x - 7)(x + 9) ## 44. Factor 9x² + 6x - **GCF:** 3x - 3x(3x + 2) ## 45. Factor: x² - 16 - **Difference of Squares:** - (x + 4)(x - 4) ## 46. What are the solutions of the equation (x - 4)(x + 3) = 0 ? - x - 4 = 0 - x = 4 - x + 3 = 0 - x = -3 - **Solutions:** x = 4 or x = -3 ## 48. Find the discriminant and the number of solutions for the equation x² + 3x - 28 = 0 - **Discriminant:** b² - 4ac - **a = 1, b = 3, c = -28** - (3)² - 4(1)(-28) - 9 + 112 - 121 - **Number of Solutions:** 2 solutions ## 50. What are the solutions of the equation 2(x+3)² - 5 = 27 - 2(x + 3)² - 5 = 27 - 2(x + 3)² = 32 - (x + 3)² = 16 - (x + 3) = ± 4 - x = 1 ## 52. What are the solutions of the equation 2x² - 5x = -6 - 2x² - 5x + 6 = 0 - **The discriminant is a negative number, therefore there are no real solutions. ** ## 54. Describe all the transformations to the parent function f(x) = x² of f(x) = -4(x - 7)² + 15 - **Reflection across the x-axis:** The negative sign in front of the 4 multiplies the output of the function by -1, reflecting it across the x-axis. - **Vertical stretch by 4:** The 4 in front of the parentheses multiplies the output of the function by 4, stretching it vertically by a factor of 4. - **Right 7:** The (x - 7) inside the parentheses shifts the graph 7 units to the right. - **Up 15:** The + 15 outside the parentheses shifts the graph 15 units upward. ## 55. Describe all the transformations to the parent function f(x) = x² of f(x) = ½x²-9 - **Vertical shrink by ½:** The ½ in front of the x² multiplies the output of the function by ½, shrinking it vertically by a factor of ½. - **Down 9:** The -9 outside the parentheses shifts the graph 9 units downward. ## 56. What is the vertex form of the quadratic function: y = x² - 3x - 10 - **Complete the square:** - y = (x² - 3x) - 10 - y = (x² - 3x + 9/4) - 10 - 9/4 - y = (x - 3/2)² - 49/4 - **Vertex form:** y = (x - 3/2)² - 49/4 ## 57. What is the standard form of the following function: y = (x + 4)² - 13 - **Expand the square:** - y = (x + 4)(x + 4) - 13 - y = x² + 4x + 4x + 16 - 13 - **Standard form:** y = x² + 8x + 3 ## For questions #58-60, use the following information to answer the following questions. - The function s(t) = -16t² + vt + h represents the height of an object, s, in feet, above the ground in relation to the time, t, in seconds since the object was thrown into the air with an initial velocity of v feet per second at an initial height of h feet. - A baseball player hits a baseball 4 feet above the ground with an initial velocity of 80 feet per second. - s(t) = -16t² + 80t + 4 ## 58. What is the ball's maximum height? - **The maximum height is the y-coordinate of the vertex of the parabola.** - **Find t of the vertex:** t = -b / 2a, where a = -16 and b = 80. - t = -80 / (2 * -16) = 2.5 seconds. - **Find the maximum height (s) by plugging in t = 2.5 into the equation:** - s(2.5) = -16(2.5)² + 80(2.5) + 4 - s(2.5) = 104 feet - **Maximum height:** 104 feet. ## 59. When will the ball reach the ground? - **The ball reaches the ground when s(t) = 0** - **Solve the quadratic equation:** -16t² + 80t + 4 = 0 using the quadratic formula. - **Solutions:** t ≈ 5.05 seconds or t ≈ -0.05 seconds. - **The negative solution is not relevant in this context.** - **Time to reach the ground:** ≈ 5.05 seconds ## 60. At what time will the ball reach its maximum height? - **The ball reaches its maximum height at the x-coordinate of the vertex.** - **t of the vertex:** t = -b / 2a, where a = -16 and b = 80. - t = -80 / (2 * -16) = 2.5 seconds - **Time to reach the maximum height:** 2.5 seconds.