answer:To solve this problem, we follow the transformations step by step: 1. **Initial Parabola**: The given parabola is y = -2x^2. This is our starting point. 2. **First Shift (1 unit to the left)**: Shifting a graph to the left by 1 unit means we replace x with (x + 1). Therefore, after the first shift, the equation of the parabola becomes: [ y = -2(x + 1)^2. ] 3. **Second Shift (3 units up)**: Shifting a graph up by 3 units means we add 3 to the y-value. Therefore, after the second shift, the equation of the parabola becomes: [ y = -2(x + 1)^2 + 3. ] Combining these steps, the analytical expression of the parabola after these two shifts is: [ boxed{y = -2(x + 1)^2 + 3}. ] Therefore, the correct answer is boxed{text{A}}.

answer:From the problem, we use the numbers m and n obtained from rolling the dice consecutively as the coordinates of point P(m, n). Analyzing the situation, we find that both m and n can take on 6 different values, so there are a total of 6 times 6 = 36 possible outcomes for point P. The condition for point P(m, n) to fall inside the circle x^2+y^2=16 is m^2+n^2<16. The pairs that satisfy this condition are (1,1), (1,2), (1,3), (2,1), (2,2), (2,3), (3,1), and (3,2), making a total of 8 such pairs. Therefore, the probability P of point P falling inside the circle is P = frac{8}{36} = frac{2}{9}. Hence, the answer is boxed{frac{2}{9}}. By analyzing the problem, we find that both m and n can take on 6 different values. Using the principle of counting, we can determine the number of possible outcomes for point P. Then, by listing the conditions under which P(m, n) falls inside the circle x^2+y^2=16, i.e., m^2+n^2<16, we can find the number of such outcomes. Using the formula for the probability of equally likely events, we can calculate the answer. This problem tests the calculation of the probability of equally likely events. The key to solving the problem is to calculate the total number of basic events and the total number of basic events that the event under study includes.

answer:1. Convert 345_{10} into base-4. Start by identifying the largest power of 4 less than 345, which is 4^3=64. 2. Calculate the highest multiple of 64 that is less than or equal to 345: 345 div 64 approx 5.39 quad text{(take the integer part)} quad 5 cdot 64 = 320 3. Compute the remainder after subtracting 320 from 345: 345 - 320 = 25 4. Continue with the next highest power of 4, 4^2=16, to handle the remainder 25: 25 div 16 = 1.56 quad text{(take the integer part)} quad 1 cdot 16 = 16 5. Compute the new remainder: 25 - 16 = 9 6. Use 4^1=4 for the next step: 9 div 4 = 2.25 quad text{(take the integer part)} quad 2 cdot 4 = 8 7. Final remainder: 9 - 8 = 1 8. Putting it all together, 345_{10} = 5 cdot 4^3 + 1 cdot 4^2 + 2 cdot 4^1 + 1 cdot 4^0 = 5121_4. 9. Count the odd digits: 5, 1, 1 (three odd digits). 10. Calculate the sum of these odd digits: 5 + 1 + 1 = 7. Conclusion: The number of odd digits in the base-4 representation of 345_{10} is boxed{3}, and the sum of these odd digits is boxed{7}.

answer:1. **Identify the coordinates and slopes**: - Coordinates given: E = (1, 1), F = (101, 21), H = (3, y). - Slope of EF: [ m_1 = frac{21 - 1}{101 - 1} = frac{20}{100} = frac{1}{5} ] 2. **Determine the slope of EH and find y**: - Slope of EH: [ m_2 = frac{y - 1}{3 - 1} = frac{y - 1}{2} ] - Since EF and EH are perpendicular, m_1 cdot m_2 = -1: [ frac{1}{5} cdot frac{y - 1}{2} = -1 implies y - 1 = -10 implies y = -9 ] 3. **Calculate the lengths of EF and EH**: - Length of EF: [ EF = sqrt{(101 - 1)^2 + (21 - 1)^2} = sqrt{100^2 + 20^2} = sqrt{10000 + 400} = sqrt{10400} = 20sqrt{26} ] - Length of EH: [ EH = sqrt{(3 - 1)^2 + (-9 - 1)^2} = sqrt{2^2 + (-10)^2} = sqrt{4 + 100} = sqrt{104} = 2sqrt{26} ] 4. **Calculate the area of rectangle EFGH**: - Area of EFGH: [ text{Area} = EF times EH = 20sqrt{26} times 2sqrt{26} = 40 times 26 = 1040 ] - The area of rectangle EFGH is 1040. The final answer is boxed{1040} (B)