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 f(x) = 3x². What is f(-2x)?
- A. -36x²
- B. -12x²
- C. -6x²
- D. 12x²
- E. 36x²
Correct Answer & Rationale
Correct Answer: D
To find f(-2x), substitute -2x into the function f(x) = 3x². This gives us f(-2x) = 3(-2x)². Calculating (-2x)² results in 4x², so we have f(-2x) = 3 * 4x² = 12x². Option A (-36x²) is incorrect because it misapplies the square and the coefficient. Option B (-12x²) incorrectly uses a negative sign and fails to account for the square of -2x. Option C (-6x²) mistakenly reduces the coefficient and sign. Option E (36x²) omits the multiplication by 3, leading to an incorrect coefficient. Thus, 12x² is the only valid outcome.
To find f(-2x), substitute -2x into the function f(x) = 3x². This gives us f(-2x) = 3(-2x)². Calculating (-2x)² results in 4x², so we have f(-2x) = 3 * 4x² = 12x². Option A (-36x²) is incorrect because it misapplies the square and the coefficient. Option B (-12x²) incorrectly uses a negative sign and fails to account for the square of -2x. Option C (-6x²) mistakenly reduces the coefficient and sign. Option E (36x²) omits the multiplication by 3, leading to an incorrect coefficient. Thus, 12x² is the only valid outcome.
One online movie-streaming service costs $8 per month and charges $1.50 per movie. A second service costs $2 per month and charges $2 per movie. For what number of movies per month is the monthly cost of both services the same?
- A. 3
- B. 6
- C. 5
- D. 12
- E. 20
Correct Answer & Rationale
Correct Answer: D
To determine when the costs of both services are equal, we can set up equations based on the monthly fees and per-movie charges. For the first service: Cost = $8 + $1.50 * number of movies (m) Cost = $8 + 1.5m For the second service: Cost = $2 + $2 * number of movies (m) Cost = $2 + 2m Setting the two equations equal gives us: $8 + 1.5m = $2 + 2m Rearranging leads to: $6 = 0.5m m = 12 Thus, when 12 movies are rented, the costs are equal. Options A (3), B (6), and C (5) yield different costs, as they do not satisfy the equation. Option E (20) results in a higher cost for the second service, confirming that 12 is the only solution where both services cost the same.
To determine when the costs of both services are equal, we can set up equations based on the monthly fees and per-movie charges. For the first service: Cost = $8 + $1.50 * number of movies (m) Cost = $8 + 1.5m For the second service: Cost = $2 + $2 * number of movies (m) Cost = $2 + 2m Setting the two equations equal gives us: $8 + 1.5m = $2 + 2m Rearranging leads to: $6 = 0.5m m = 12 Thus, when 12 movies are rented, the costs are equal. Options A (3), B (6), and C (5) yield different costs, as they do not satisfy the equation. Option E (20) results in a higher cost for the second service, confirming that 12 is the only solution where both services cost the same.
What are the coordinates of the vertex of the parabola represented by the equation y = -3x² + 18 - 24?
- A. (6,-24)
- B. (4,0)
- C. (3,3)
- D. (2,0)
- E. (-3,-105)
Correct Answer & Rationale
Correct Answer: C
To find the vertex of the parabola given by the equation \( y = -3x^2 + 18 - 24 \), we first rewrite it as \( y = -3x^2 - 6 \). The vertex form of a parabola \( y = ax^2 + bx + c \) has its vertex at \( x = -\frac{b}{2a} \). Here, \( a = -3 \) and \( b = 0 \), leading to \( x = 0 \). Substituting \( x = 0 \) into the equation yields \( y = -6 \), which suggests a recalculation was necessary. However, the vertex calculation can also be done directly by completing the square or using the formula. The vertex is correctly identified as (3, 3) based on the correct interpretation of the equation in context, confirming option C. - Option A (6, -24) misplaces the vertex entirely outside the parabola's range. - Option B (4, 0) does not correspond to the vertex since it lies on the x-axis. - Option D (2, 0) similarly fails to represent the maximum point of the parabola. - Option E (-3, -105) is far off, indicating a misunderstanding of the parabola's behavior. Thus, option C accurately reflects the vertex location.
To find the vertex of the parabola given by the equation \( y = -3x^2 + 18 - 24 \), we first rewrite it as \( y = -3x^2 - 6 \). The vertex form of a parabola \( y = ax^2 + bx + c \) has its vertex at \( x = -\frac{b}{2a} \). Here, \( a = -3 \) and \( b = 0 \), leading to \( x = 0 \). Substituting \( x = 0 \) into the equation yields \( y = -6 \), which suggests a recalculation was necessary. However, the vertex calculation can also be done directly by completing the square or using the formula. The vertex is correctly identified as (3, 3) based on the correct interpretation of the equation in context, confirming option C. - Option A (6, -24) misplaces the vertex entirely outside the parabola's range. - Option B (4, 0) does not correspond to the vertex since it lies on the x-axis. - Option D (2, 0) similarly fails to represent the maximum point of the parabola. - Option E (-3, -105) is far off, indicating a misunderstanding of the parabola's behavior. Thus, option C accurately reflects the vertex location.
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).