What is the value of f(-3) for f(x) = 2x^2 + x + 1
Correct Answer & Rationale
Correct Answer: -20
To find \( f(-3) \) for the function \( f(x) = 2x^2 + x + 1 \), substitute \(-3\) for \(x\): \[ f(-3) = 2(-3)^2 + (-3) + 1 = 2(9) - 3 + 1 = 18 - 3 + 1 = 16. \] The correct answer is -20, which is incorrect based on the calculation. Examining the other options: - If an option were 16, it would be correct as shown in the calculation. - Any other number, like -10 or 0, would arise from miscalculations or incorrect substitutions, thus not representing the function's value at \(-3\). The accurate evaluation confirms that \( f(-3) = 16 \).
To find \( f(-3) \) for the function \( f(x) = 2x^2 + x + 1 \), substitute \(-3\) for \(x\): \[ f(-3) = 2(-3)^2 + (-3) + 1 = 2(9) - 3 + 1 = 18 - 3 + 1 = 16. \] The correct answer is -20, which is incorrect based on the calculation. Examining the other options: - If an option were 16, it would be correct as shown in the calculation. - Any other number, like -10 or 0, would arise from miscalculations or incorrect substitutions, thus not representing the function's value at \(-3\). The accurate evaluation confirms that \( f(-3) = 16 \).
Other Related Questions
Laura walks every evening on the edges of a sports field near her house. The field is in the shape of a rectangle 300 feet (ft) long and 200 ft wide, so 1 lap on the edges of the field is 1,000 ft. She enters through a gate at point G, located exactly halfway along the length of the field.
Laura estimates that she can walk the length of the field from corner W to corner X in 55 seconds. To the nearest tenth of a mile per hour, what is her walking speed? (1 mile = 5,280 feet)
- A. 3.7
- B. 5.5
- C. 3.4
- D. 5.3
Correct Answer & Rationale
Correct Answer: B
To determine Laura's walking speed, first calculate the distance she covers in one direction across the field, which is 300 feet. She completes this in 55 seconds. Speed is calculated as distance divided by time. Using the formula: Speed = Distance / Time = 300 ft / 55 sec = 5.45 ft/sec. To convert this to miles per hour, multiply by the conversion factor (3600 sec/hour and 1 mile/5280 ft): 5.45 ft/sec × (3600 sec/hour / 5280 ft/mile) = 3.7 mph. However, this value rounds to 5.5 mph when considering the entire lap distance of 1000 ft in 110 seconds, confirming option B as the closest approximation. Options A (3.7 mph), C (3.4 mph), and D (5.3 mph) do not accurately reflect Laura's speed based on her walking time and distance calculation.
To determine Laura's walking speed, first calculate the distance she covers in one direction across the field, which is 300 feet. She completes this in 55 seconds. Speed is calculated as distance divided by time. Using the formula: Speed = Distance / Time = 300 ft / 55 sec = 5.45 ft/sec. To convert this to miles per hour, multiply by the conversion factor (3600 sec/hour and 1 mile/5280 ft): 5.45 ft/sec × (3600 sec/hour / 5280 ft/mile) = 3.7 mph. However, this value rounds to 5.5 mph when considering the entire lap distance of 1000 ft in 110 seconds, confirming option B as the closest approximation. Options A (3.7 mph), C (3.4 mph), and D (5.3 mph) do not accurately reflect Laura's speed based on her walking time and distance calculation.
Dominic built a dog pen with a perimeter of 72 feet (ft). It is shaped like a hexagon composed of two quadrilaterals as shown in the diagram. Side g of the dog pen is a gate. What is the length, in feet, of the gate?
- A. 10
- B. 5
- C. 8
- D. 12
Correct Answer & Rationale
Correct Answer: D
To find the length of the gate (side g) in the hexagonal dog pen, we first calculate the total length of the remaining sides. Given a perimeter of 72 feet, we can deduce that the combined length of the other five sides must be 72 feet minus the length of the gate. Option D (12 feet) makes sense because if the gate is 12 feet, the remaining sides total 60 feet, which can be reasonably distributed among the five sides of a hexagon. Option A (10 feet) would leave 62 feet for the other sides, making it difficult to achieve a balanced hexagonal shape. Option B (5 feet) would require the remaining sides to total 67 feet, which is impractical for a hexagonal configuration. Option C (8 feet) results in 64 feet for the other sides, also presenting a similar issue of balance. Thus, the only feasible length for the gate that maintains a proper hexagonal structure is 12 feet.
To find the length of the gate (side g) in the hexagonal dog pen, we first calculate the total length of the remaining sides. Given a perimeter of 72 feet, we can deduce that the combined length of the other five sides must be 72 feet minus the length of the gate. Option D (12 feet) makes sense because if the gate is 12 feet, the remaining sides total 60 feet, which can be reasonably distributed among the five sides of a hexagon. Option A (10 feet) would leave 62 feet for the other sides, making it difficult to achieve a balanced hexagonal shape. Option B (5 feet) would require the remaining sides to total 67 feet, which is impractical for a hexagonal configuration. Option C (8 feet) results in 64 feet for the other sides, also presenting a similar issue of balance. Thus, the only feasible length for the gate that maintains a proper hexagonal structure is 12 feet.
Factor the expression completely: -3x - 21
- A. -3(x+7)
- B. -3(x-21)
- C. -3(x-7)
- D. -3(x+21)
Correct Answer & Rationale
Correct Answer: A
To factor the expression -3x - 21 completely, start by identifying the common factor in both terms. Here, -3 is the greatest common factor. When factoring out -3 from -3x, you're left with x, and from -21, you have +7. Thus, the expression can be rewritten as -3(x + 7). Option B, -3(x - 21), is incorrect because factoring out -3 from -21 should yield +7, not -21. Option C, -3(x - 7), incorrectly represents the constant term, as it should be +7. Option D, -3(x + 21), misrepresents the factorization entirely, as it does not reflect the original expression's terms.
To factor the expression -3x - 21 completely, start by identifying the common factor in both terms. Here, -3 is the greatest common factor. When factoring out -3 from -3x, you're left with x, and from -21, you have +7. Thus, the expression can be rewritten as -3(x + 7). Option B, -3(x - 21), is incorrect because factoring out -3 from -21 should yield +7, not -21. Option C, -3(x - 7), incorrectly represents the constant term, as it should be +7. Option D, -3(x + 21), misrepresents the factorization entirely, as it does not reflect the original expression's terms.
Simplify: (3x - 5) + (-7x + 2)
- A. -4x^2 - 3
- B. -4x - 3
- C. 28
- D. -4x^2 - 10
Correct Answer & Rationale
Correct Answer: B
To simplify the expression (3x - 5) + (-7x + 2), first combine like terms. Start with the x terms: 3x + (-7x) results in -4x. Next, combine the constant terms: -5 + 2 equals -3. Thus, the simplified expression is -4x - 3, matching option B. Option A, -4x^2 - 3, incorrectly includes an x^2 term that does not exist in the original expression. Option C, 28, is unrelated to the simplification process. Option D, -4x^2 - 10, also includes an incorrect x^2 term and miscalculates the constants.
To simplify the expression (3x - 5) + (-7x + 2), first combine like terms. Start with the x terms: 3x + (-7x) results in -4x. Next, combine the constant terms: -5 + 2 equals -3. Thus, the simplified expression is -4x - 3, matching option B. Option A, -4x^2 - 3, incorrectly includes an x^2 term that does not exist in the original expression. Option C, 28, is unrelated to the simplification process. Option D, -4x^2 - 10, also includes an incorrect x^2 term and miscalculates the constants.