Rise Placement Test Functions and Quadratic Equations Questions and Answers — Questions and Answers
Question 1: Given the function f(x) = 2x² - 3x + 4, what is the value of f(-3)?
- 7
- -5
- 31 (Correct answer)
- 13
Correct answer: 31
To find f(-3), substitute -3 for every instance of x in the function: f(-3) = 2(-3)² - 3(-3) + 4. Following the order of operations, first calculate the exponent: (-3)² = 9. Then, perform the multiplication: 2(9) = 18 and -3(-3) = 9. Finally, perform the addition: 18 + 9 + 4 = 31.
Question 2: What is the domain of the function g(x) = √(x - 7)?
- All real numbers
- x > 7
- x ≤ 7
- x ≥ 7 (Correct answer)
Correct answer: x ≥ 7
The expression inside a square root cannot be negative for the function to yield a real number. Therefore, we must set the expression (x - 7) to be greater than or equal to zero. The inequality is x - 7 ≥ 0. Adding 7 to both sides gives x ≥ 7. In interval notation, this is [7, ∞).
Question 3: A rocket is launched into the air, and its height in feet at time t (in seconds) is given by the function h(t) = -16t² + 96t. What is the maximum height the rocket reaches?
- 96 feet
- 144 feet (Correct answer)
- 3 feet
- 160 feet
Correct answer: 144 feet
The graph of this quadratic function is a downward-opening parabola. The maximum height occurs at the vertex. To find the time (t-coordinate of the vertex), use the formula t = -b / (2a). Here, a = -16 and b = 96. So, t = -96 / (2 * -16) = -96 / -32 = 3 seconds. To find the maximum height, substitute this time back into the function: h(3) = -16(3)² + 96(3) = -16(9) + 288 = -144 + 288 = 144 feet.
Question 4: Which of the following are the solutions to the quadratic equation 2x² - 5x - 4 = 0?
- x = (5 ± √57) / 4 (Correct answer)
- x = (-5 ± √57) / 4
- x = (5 ± √7) / 4
- x = (-5 ± √7) / 2
Correct answer: x = (5 ± √57) / 4
This quadratic equation does not factor easily, so the quadratic formula, x = [-b ± √(b² - 4ac)] / 2a, is the best method. For this equation, a = 2, b = -5, and c = -4. Substituting these values gives: x = [ -(-5) ± √((-5)² - 4(2)(-4)) ] / (2*2) = [ 5 ± √(25 + 32) ] / 4 = (5 ± √57) / 4.
Question 5: If f(x) = 3x - 2 and g(x) = x² + 1, what is the composite function (f ∘ g)(x), which is equivalent to f(g(x))?
- 9x² - 12x + 5
- 3x³ - 2x² + 3x - 2
- 3x² + 1
- 3x² - 1 (Correct answer)
Correct answer: 3x² - 1
To find f(g(x)), we substitute the entire function g(x) into f(x) wherever we see x. So, f(g(x)) = f(x² + 1). Now, we apply the rule for f(x), which is 3*(input) - 2. Our input is (x² + 1). This gives us 3(x² + 1) - 2. Distributing the 3 gives 3x² + 3 - 2, which simplifies to 3x² - 1.
Question 6: What are the coordinates of the vertex of the parabola defined by the equation y = 2x² + 12x + 10?
- (3, 64)
- (-3, -8) (Correct answer)
- (-6, 10)
- (3, -10)
Correct answer: (-3, -8)
The x-coordinate of the vertex of a parabola in the form y = ax² + bx + c is found using the formula x = -b / (2a). In this equation, a = 2 and b = 12. So, x = -12 / (2 * 2) = -12 / 4 = -3. To find the y-coordinate, substitute x = -3 back into the original equation: y = 2(-3)² + 12(-3) + 10 = 2(9) - 36 + 10 = 18 - 36 + 10 = -8. Therefore, the vertex is at (-3, -8).
Rise Placement Test Functions and Quadratic Equations Questions and Answers
Given the function f(x) = 2x² - 3x + 4, what is the value of f(-3)?