1.
Which shows the quadratic function expressed in factored form? f(x) = 2x^{2} + 9x – 5
Correct Answer
B. (2x – 1)(x + 5)
Explanation
The given quadratic function f(x) = 2x^2 + 9x - 5 can be expressed in factored form as (2x - 1)(x + 5).
2.
The height of a crab h(t) in meters, dropped by a seagull to the rocks below, is given by the function h(t) = –5t^{2} + 20, where t is the time, in seconds, after it is released. What are the zeros of the function?
Correct Answer
C. (–2, 0) and (20, 0)
Explanation
The zeros of a function are the values of the independent variable that make the function equal to zero. In this case, we need to find the values of t that make h(t) = 0.
Substituting h(t) = 0 into the given function, we get:
0 = -5t^2 + 20
Rearranging the equation, we have:
5t^2 = 20
Dividing both sides by 5, we get:
t^2 = 4
Taking the square root of both sides, we have:
t = ±2
Therefore, the zeros of the function are t = -2 and t = 2. However, the question asks for the zeros in terms of coordinates. Since the function is h(t), the zeros can be written as (-2, 0) and (2, 0).
Therefore, the correct answer is (–2, 0) and (20, 0).
3.
What is the axis of symmetry for the function –x^{2} + 7x + 30
Correct Answer
B. X = 3.5
Explanation
The axis of symmetry for a quadratic function is given by the formula x = -b/2a. In this case, the quadratic function is -x^2 + 7x + 30. By comparing with the general form ax^2 + bx + c, we can see that a = -1 and b = 7. Plugging these values into the formula, we get x = -7/(2*(-1)) = 3.5. Therefore, the axis of symmetry for the function is x = 3.5.
4.
A volleyball is hit straight upward. The graph shows its height, h(t), in meters, at time t. Which is the function in a factored form that estimates the height of the volleyball at any given time?
Correct Answer
D. –5x(x – 2)
Explanation
The factored form of the function that estimates the height of the volleyball at any given time is –5x(x – 2). This can be determined by analyzing the given options and comparing them to the graph of the height function. The function –5x(x – 2) is a quadratic function in factored form, where x represents time and (x – 2) represents the time at which the volleyball reaches its maximum height. The coefficient -5 indicates that the height decreases over time.
5.
What is the y–intercept of the quadratic function f(x) = (2x + 4)(x – 3)?
Correct Answer
A. (0, –12)
Explanation
The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis. To find the y-intercept, we set x=0 and evaluate the function. When we substitute x=0 into the function f(x) = (2x + 4)(x – 3), we get f(0) = (2(0) + 4)(0 – 3) = (0 + 4)(0 – 3) = 4(-3) = -12. Therefore, the y-intercept of the quadratic function is (0, -12).
6.
A rocket is shot into the air. The height of the rocket is modeled by the function h(t) = –5t^{2} + 45t, where h(t) is the height in meters and t is the time in seconds. When will the rocket hit the ground?
Correct Answer
C. 9 sec
Explanation
The rocket will hit the ground when its height is equal to zero. To find the time when this happens, we need to solve the equation h(t) = 0. Substituting the given function, we get -5t^2 + 45t = 0. Factoring out -5t, we have -5t(t - 9) = 0. This equation is true when either -5t = 0 or t - 9 = 0. Solving these equations, we find t = 0 or t = 9. Since time cannot be negative in this context, the rocket will hit the ground after 9 seconds.
7.
What are the coordinates of the vertex of f(x) = 2x^{2} – 6x – 20?
Correct Answer
B. (1.5, –24.5)
Explanation
The given quadratic function is in the form f(x) = ax^2 + bx + c. The vertex of a quadratic function can be found using the formula x = -b/2a. In this case, a = 2 and b = -6. Plugging these values into the formula, we get x = -(-6)/(2*2) = 6/4 = 1.5. To find the y-coordinate of the vertex, we substitute this value of x back into the function: f(1.5) = 2(1.5)^2 - 6(1.5) - 20 = -24.5. Therefore, the coordinates of the vertex are (1.5, -24.5).
8.
A rock is thrown into the river from a bridge that is 20 meters high. The table shows its height, h(t), at time t. Determine a function, in factored form, that estimates the height of the rock at any given time.
Correct Answer
D. –5(x+1)(x–4)
Explanation
The given correct answer, –5(x+1)(x–4), represents a function in factored form that estimates the height of the rock at any given time. The function is in the form of a quadratic equation, where (x+1) and (x–4) represent the factors that determine the time at which the height is being estimated. The coefficient -5 indicates that the height is decreasing over time. By plugging in different values for x, the function can be used to estimate the height of the rock at any given time.
9.
Which is the graph of the function g(x) = (2x – 1)(x – 5)?
Correct Answer
A. Graph A
Explanation
The graph of the function g(x) = (2x – 1)(x – 5) is represented by Graph A. This can be determined by analyzing the factors of the function. The factor (2x - 1) indicates that the graph will have a root at x = 1/2, and the factor (x - 5) indicates a root at x = 5. Therefore, the graph will intersect the x-axis at these points. Additionally, the coefficient 2 in front of (2x - 1) indicates that the graph will be steeper compared to the other graphs. Hence, Graph A is the correct representation of the given function.
10.
A fountain shoots water from a muzzle at its top. The function f(x) = –5t^{2} + 10t + 15 describes the height of the water h(t) in meters, t seconds after it leaves the nozzle. What is the maximum height of the water spout?
Correct Answer
D. 20m
Explanation
The given function represents a quadratic equation in the form of h(t) = -5t^2 + 10t + 15. This equation represents the height of the water spout at any given time t. The coefficient of the t^2 term is negative, indicating that the graph of the equation is a downward-opening parabola. The maximum height of the water spout will occur at the vertex of this parabola. To find the vertex, we can use the formula t = -b/2a, where a = -5 and b = 10. Plugging in these values, we get t = -10/(-10) = 1. Substituting t = 1 into the equation, we get h(1) = -5(1)^2 + 10(1) + 15 = 20. Therefore, the maximum height of the water spout is 20m.