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.
Other Related Questions
Which of the following expressions is equivalent to (4x²)(5x³)?
- A. 9xâµ
- B. 9xâ¶
- C. 20xâµ
- D. 20xâ¶
- E. 20xâ¹
Correct Answer & Rationale
Correct Answer: C
To find the equivalent expression for (4x²)(5x³), multiply the coefficients (4 and 5) and add the exponents of x (2 and 3). Thus, 4 × 5 equals 20, and x² × x³ results in x^(2+3) = x⁵. This gives us 20x⁵. Option A (9x⁶) is incorrect because it miscalculates both the coefficient and the exponent. Option B (9x⁷) also miscalculates both the coefficient and exponent. Option D (20x⁶) correctly identifies the coefficient but incorrectly adds the exponents. Option E (20x¹) miscalculates the exponent entirely. Only option C accurately represents the expression as 20x⁵.
To find the equivalent expression for (4x²)(5x³), multiply the coefficients (4 and 5) and add the exponents of x (2 and 3). Thus, 4 × 5 equals 20, and x² × x³ results in x^(2+3) = x⁵. This gives us 20x⁵. Option A (9x⁶) is incorrect because it miscalculates both the coefficient and the exponent. Option B (9x⁷) also miscalculates both the coefficient and exponent. Option D (20x⁶) correctly identifies the coefficient but incorrectly adds the exponents. Option E (20x¹) miscalculates the exponent entirely. Only option C accurately represents the expression as 20x⁵.
A medium-sized grain of sand can be approximated as a cube with an edge length of 5×10â»â´ meters. Which expression best represents the number of medium-sized sand grains that could be lined up side by side to result in a total length of 1 meter?
- A. 2×10³
- B. 2×10â´
- C. 2×10âµ
- D. 5×10³
- E. 5×10â´
Correct Answer & Rationale
Correct Answer: B
To determine how many medium-sized sand grains can be lined up to equal 1 meter, we first calculate the volume of one grain, approximated as a cube with an edge length of 5×10⁻⁴ meters. The length of one grain is 5×10⁻⁴ meters. To find the number of grains in 1 meter, divide 1 meter (1×10⁰) by the length of one grain: 1×10⁰ / 5×10⁻⁴ = 2×10³. Thus, option B (2×10³) accurately represents the number of grains. Options A (2×10³) and D (5×10³) are incorrect due to miscalculating the division. Option C (2×10⁻) and E (5×10⁵) misrepresent the scale entirely, either by underestimating or overestimating the number of grains.
To determine how many medium-sized sand grains can be lined up to equal 1 meter, we first calculate the volume of one grain, approximated as a cube with an edge length of 5×10⁻⁴ meters. The length of one grain is 5×10⁻⁴ meters. To find the number of grains in 1 meter, divide 1 meter (1×10⁰) by the length of one grain: 1×10⁰ / 5×10⁻⁴ = 2×10³. Thus, option B (2×10³) accurately represents the number of grains. Options A (2×10³) and D (5×10³) are incorrect due to miscalculating the division. Option C (2×10⁻) and E (5×10⁵) misrepresent the scale entirely, either by underestimating or overestimating the number of grains.
When Henry plays the songs on the playlist in a random order, what is the probability a rock song will be played first?
- A. 3/4
- B. 1/3
- C. 1/4
- D. 3/10
- E. 5/16
Correct Answer & Rationale
Correct Answer: D
To find the probability of a rock song being played first, we need to know the total number of songs and how many of those are rock songs. If there are 3 rock songs and a total of 10 songs, the probability is calculated as the number of favorable outcomes (rock songs) divided by the total outcomes (all songs). Thus, the probability is 3/10, which corresponds to option D. Option A (3/4) overestimates the likelihood by implying a much higher proportion of rock songs. Option B (1/3) incorrectly assumes there are fewer total songs than there actually are. Option C (1/4) underrepresents the rock songs available. Option E (5/16) is irrelevant as it does not align with the total number of songs.
To find the probability of a rock song being played first, we need to know the total number of songs and how many of those are rock songs. If there are 3 rock songs and a total of 10 songs, the probability is calculated as the number of favorable outcomes (rock songs) divided by the total outcomes (all songs). Thus, the probability is 3/10, which corresponds to option D. Option A (3/4) overestimates the likelihood by implying a much higher proportion of rock songs. Option B (1/3) incorrectly assumes there are fewer total songs than there actually are. Option C (1/4) underrepresents the rock songs available. Option E (5/16) is irrelevant as it does not align with the total number of songs.
Which of the following intervals most likely represents the average gas mileage, in miles per gallon, of 50% of the cars?
- A. 20 to 32
- B. 24 to 32
- C. 29 to 32
- D. 30 to 44
- E. 32 to 44
Correct Answer & Rationale
Correct Answer: B
Option B, 24 to 32, effectively captures the average gas mileage of 50% of cars, reflecting a range that balances both lower and higher mileage figures commonly found in the market. Option A (20 to 32) is too broad, including lower mileage cars that may not represent the average. Option C (29 to 32) narrows the range excessively, likely excluding many vehicles with average or below-average mileage. Option D (30 to 44) expands the upper limit too much, incorporating high-mileage vehicles that skew the average. Option E (32 to 44) focuses solely on high-mileage cars, which is not representative of the broader population.
Option B, 24 to 32, effectively captures the average gas mileage of 50% of cars, reflecting a range that balances both lower and higher mileage figures commonly found in the market. Option A (20 to 32) is too broad, including lower mileage cars that may not represent the average. Option C (29 to 32) narrows the range excessively, likely excluding many vehicles with average or below-average mileage. Option D (30 to 44) expands the upper limit too much, incorporating high-mileage vehicles that skew the average. Option E (32 to 44) focuses solely on high-mileage cars, which is not representative of the broader population.