Through which pair of points could a line of best fit be drawn for the data on the scatterplot?
- A. (0, 36) and (11, 74)
- B. (1, 39) and (6, 60)
- C. (5, 50) and (6, 60)
- D. (6, 60) and (8, 60)
- E. (8, 60) and (11, 74)
Correct Answer & Rationale
Correct Answer: A
Option A, with points (0, 36) and (11, 74), shows a significant range in both x and y values, indicating a strong upward trend that aligns well with the overall direction of the data. Option B, while showing an upward trend, has a narrower range and may not represent the overall data as effectively. Option C features two points that are too close together, limiting their ability to define a clear line of best fit. Option D includes points with the same y-value, suggesting a horizontal line that does not capture the data's trend. Option E, like A, has a valid upward trend but does not span the data range as effectively as A.
Option A, with points (0, 36) and (11, 74), shows a significant range in both x and y values, indicating a strong upward trend that aligns well with the overall direction of the data. Option B, while showing an upward trend, has a narrower range and may not represent the overall data as effectively. Option C features two points that are too close together, limiting their ability to define a clear line of best fit. Option D includes points with the same y-value, suggesting a horizontal line that does not capture the data's trend. Option E, like A, has a valid upward trend but does not span the data range as effectively as A.
Other Related Questions
Let g(x) = x². What is the average rate of change of the function from x = 4 to x = 8?
- A. 1/12
- B. $2
- C. $4
- D. $12
- E. $48
Correct Answer & Rationale
Correct Answer: C
To determine the average rate of change of the function g(x) = x² from x = 4 to x = 8, we use the formula: (g(b) - g(a)) / (b - a), where a = 4 and b = 8. Calculating g(4) = 4² = 16 and g(8) = 8² = 64. Thus, the average rate of change is (64 - 16) / (8 - 4) = 48 / 4 = 12. Option A (1/12) is incorrect as it underestimates the change. Option B ($2) and Option D ($12) miscalculate the average rate. Option E ($48) represents the total change but does not account for the interval length. The correct average rate of change is $12, reflecting the consistent increase of the function over the specified interval.
To determine the average rate of change of the function g(x) = x² from x = 4 to x = 8, we use the formula: (g(b) - g(a)) / (b - a), where a = 4 and b = 8. Calculating g(4) = 4² = 16 and g(8) = 8² = 64. Thus, the average rate of change is (64 - 16) / (8 - 4) = 48 / 4 = 12. Option A (1/12) is incorrect as it underestimates the change. Option B ($2) and Option D ($12) miscalculate the average rate. Option E ($48) represents the total change but does not account for the interval length. The correct average rate of change is $12, reflecting the consistent increase of the function over the specified interval.
Square PQRS, with a side length of 5 units, will be translated 2 units to the right and 2 units up in the standard (x, y) coordinate plane. What is the area, in square units, of the image of PQRS?
- A. 20
- B. 25
- C. 40
- D. 50
- E. 100
Correct Answer & Rationale
Correct Answer: B
The area of a square is calculated by squaring the length of its sides. For square PQRS, with a side length of 5 units, the area is \(5 \times 5 = 25\) square units. Translating the square 2 units to the right and 2 units up does not alter its dimensions or area; it simply changes its position on the coordinate plane. Options A (20), C (40), D (50), and E (100) suggest changes in area due to incorrect assumptions about the effects of translation or miscalculations. The area remains constant at 25 square units, confirming option B as the only accurate choice.
The area of a square is calculated by squaring the length of its sides. For square PQRS, with a side length of 5 units, the area is \(5 \times 5 = 25\) square units. Translating the square 2 units to the right and 2 units up does not alter its dimensions or area; it simply changes its position on the coordinate plane. Options A (20), C (40), D (50), and E (100) suggest changes in area due to incorrect assumptions about the effects of translation or miscalculations. The area remains constant at 25 square units, confirming option B as the only accurate choice.
Which of the following statements is true about the graphs of f(x) = x and g(x) = 3x in the standard (x, y) coordinate plane?
- A. The graphs will not intersect.
- B. The graphs will intersect only at the point (0,0).
- C. The graphs will intersect only at the point (0,1).
- D. The graphs will intersect only at the point (1,1).
- E. The graphs will intersect only at the point (3,3).
Correct Answer & Rationale
Correct Answer: D
The graphs of f(x) = x and g(x) = 3x represent two linear functions with different slopes. The first function has a slope of 1, while the second has a slope of 3. They will intersect where their outputs are equal, which occurs when x = 1, resulting in the point (1,1). Option A is incorrect as the lines, being linear, will intersect at some point. Option B is misleading; they intersect at (0,0) but also at (1,1). Option C is false because g(1) = 3, not 1. Option E is incorrect since g(3) = 9, not 3. Thus, the only valid intersection point is (1,1).
The graphs of f(x) = x and g(x) = 3x represent two linear functions with different slopes. The first function has a slope of 1, while the second has a slope of 3. They will intersect where their outputs are equal, which occurs when x = 1, resulting in the point (1,1). Option A is incorrect as the lines, being linear, will intersect at some point. Option B is misleading; they intersect at (0,0) but also at (1,1). Option C is false because g(1) = 3, not 1. Option E is incorrect since g(3) = 9, not 3. Thus, the only valid intersection point is (1,1).
What is the product of the two polynomials: (x - 5)(x² - 3x + 6)?
- A. x³ - 8x² + 21x - 30
- B. x³ - 8x² - 21x - 30
- C. x³ - 8x² - 9x - 30
- D. x³ + 8x² + 21x + 30
- E. x³ + 8x² - 9x + 30
Correct Answer & Rationale
Correct Answer: A
To find the product of the polynomials (x - 5)(x² - 3x + 6), we apply the distributive property (FOIL method). 1. Multiply x by each term in the second polynomial: - x * x² = x³ - x * (-3x) = -3x² - x * 6 = 6x 2. Multiply -5 by each term in the second polynomial: - -5 * x² = -5x² - -5 * (-3x) = 15x - -5 * 6 = -30 Combining these results yields: x³ + (-3x² - 5x²) + (6x + 15x) - 30 = x³ - 8x² + 21x - 30. Option A matches this result. Options B and C have incorrect signs for the x terms. Option D has incorrect signs for all terms, and option E has incorrect signs for the x² and x terms. Thus, only option A accurately represents the product of the polynomials.
To find the product of the polynomials (x - 5)(x² - 3x + 6), we apply the distributive property (FOIL method). 1. Multiply x by each term in the second polynomial: - x * x² = x³ - x * (-3x) = -3x² - x * 6 = 6x 2. Multiply -5 by each term in the second polynomial: - -5 * x² = -5x² - -5 * (-3x) = 15x - -5 * 6 = -30 Combining these results yields: x³ + (-3x² - 5x²) + (6x + 15x) - 30 = x³ - 8x² + 21x - 30. Option A matches this result. Options B and C have incorrect signs for the x terms. Option D has incorrect signs for all terms, and option E has incorrect signs for the x² and x terms. Thus, only option A accurately represents the product of the polynomials.