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
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.
The distance, d, in feet, it takes to come to a complete stop when driving a car r miles per hour can be found using the equation d = 1/20(r^2)+ r. If it takes a car 240 feet to come to a complete stop, what was the speed of the car, in miles per hour, when the driver began to stop it?
- A. 40
- B. 30
- C. 60
- D. 80
Correct Answer & Rationale
Correct Answer: A
To find the speed of the car when it takes 240 feet to stop, substitute d = 240 into the equation d = 1/20(r^2) + r. This leads to the equation 240 = 1/20(r^2) + r. Multiplying through by 20 simplifies to 4800 = r^2 + 20r, which rearranges to r^2 + 20r - 4800 = 0. Solving this quadratic equation yields r = 40 or r = -120. Since speed cannot be negative, the valid solution is 40 mph. Option B (30) does not satisfy the equation, leading to a shorter stopping distance. Option C (60) results in a stopping distance of 480 feet, which exceeds 240 feet. Option D (80) produces a stopping distance of 800 feet, also incorrect. Thus, only 40 mph meets the criteria.
To find the speed of the car when it takes 240 feet to stop, substitute d = 240 into the equation d = 1/20(r^2) + r. This leads to the equation 240 = 1/20(r^2) + r. Multiplying through by 20 simplifies to 4800 = r^2 + 20r, which rearranges to r^2 + 20r - 4800 = 0. Solving this quadratic equation yields r = 40 or r = -120. Since speed cannot be negative, the valid solution is 40 mph. Option B (30) does not satisfy the equation, leading to a shorter stopping distance. Option C (60) results in a stopping distance of 480 feet, which exceeds 240 feet. Option D (80) produces a stopping distance of 800 feet, also incorrect. Thus, only 40 mph meets the criteria.
Compare the zeros of function P and function Q. Which statement about the zeros of the functions is true?
- A. Function P has the greater zero, which is 9.
- B. Function P has the greater zero, which is 1.
- C. Function Q has the greater zero, which is 5.
- D. Function Q has the greater zero, which is 4.
Correct Answer & Rationale
Correct Answer: C
To determine which statement is true regarding the zeros of functions P and Q, it's essential to analyze the values given. Option A claims that function P's greater zero is 9; however, this contradicts the provided information, as 9 is not a zero for P. Option B asserts that function P's greater zero is 1, which is also incorrect if 1 is not the highest zero of P. Option D states that function Q's greater zero is 4, but if Q's zeros are higher, this option cannot be true. In contrast, option C correctly identifies that function Q has a greater zero, specifically 5, which aligns with the provided data about the functions' zeros.
To determine which statement is true regarding the zeros of functions P and Q, it's essential to analyze the values given. Option A claims that function P's greater zero is 9; however, this contradicts the provided information, as 9 is not a zero for P. Option B asserts that function P's greater zero is 1, which is also incorrect if 1 is not the highest zero of P. Option D states that function Q's greater zero is 4, but if Q's zeros are higher, this option cannot be true. In contrast, option C correctly identifies that function Q has a greater zero, specifically 5, which aligns with the provided data about the functions' zeros.
A scale drawing of a truck has a length of 3 inches (in.), as shown below. The actual truck has a length of 18 feet (ft). What scale was used for the drawing?
- A. 6 in. = 1 ft
- B. 1 in. = 15 ft
- C. 1 in. = 6 ft
- D. 15 in. = 1 ft
Correct Answer & Rationale
Correct Answer: C
To determine the scale used for the drawing, we first convert the actual truck length from feet to inches. Since 1 foot equals 12 inches, an 18-foot truck is 216 inches long (18 ft x 12 in/ft). The scale drawing shows a length of 3 inches. To find the scale, we set up the ratio of the drawing length to the actual length: 3 in. (drawing) to 216 in. (actual). Simplifying this gives us a scale of 1 in. = 72 in., which translates to 1 in. = 6 ft (since 72 in. รท 12 in/ft = 6 ft). Option A (6 in. = 1 ft) is incorrect; it implies a much larger drawing. Option B (1 in. = 15 ft) underestimates the actual size. Option D (15 in. = 1 ft) greatly exaggerates the scale, making the drawing too small.
To determine the scale used for the drawing, we first convert the actual truck length from feet to inches. Since 1 foot equals 12 inches, an 18-foot truck is 216 inches long (18 ft x 12 in/ft). The scale drawing shows a length of 3 inches. To find the scale, we set up the ratio of the drawing length to the actual length: 3 in. (drawing) to 216 in. (actual). Simplifying this gives us a scale of 1 in. = 72 in., which translates to 1 in. = 6 ft (since 72 in. รท 12 in/ft = 6 ft). Option A (6 in. = 1 ft) is incorrect; it implies a much larger drawing. Option B (1 in. = 15 ft) underestimates the actual size. Option D (15 in. = 1 ft) greatly exaggerates the scale, making the drawing too small.