1. The area of a rectangle A in square meters is modeled by the quadratic function A(w)=w(16−w), where w represents the width of the rectangle in meters. The graph of A(w) is given below. How long is the width so that the area of the rectangle is at its maximum?
A. 8 meters
B. 64 meters
C. 16 meters
D. 32 meters
2. . The path of an athlete's long jump is modeled by a function -1/4d^2+d
where d and h denote the distance and the height in meters respectively. The graph of h(d) is given by: Based on the function h(d), determine the horizontal length and the maximum height of this jump.
A. The horizontal length of the jump is 2 meters and the maximum height is 1/2 meter.
B. The horizontal length of the jump is 2 meters and the maximum height is 1 meter.
C. The horizontal length of the jump is 4 meters and the maximum height is 1/4 meter.
D. The horizontal length of the jump is 4 meters and the maximum height is 1 meter.
3. Use technology to graph the function f(x)=1/2x^2−2x . Then select the true statement below.
A. The vertex and the y-intercept of f(x) are at the same point.
B. The two zeros of f(x) are at the same point
C. The vertex and one of the zeros of f(x) are at the same point.
D. The y-intercept and one of the zeros of f(x) are at the same point.
4. The y-intercept and one of the zeros of f(x) are at the same point. h(t)=−2t^2+6t−4 where h is the height of the dolphin relative to the water level in meters at time t seconds. The graph of h(t) is given below. Then select the true statement.
A. The vertex of h(t) is at (3/2, 1/2) .
B. The two zeros of h(t) are at (−1, 0) and (−2, 0) .
C. The y-intercept of of h(t) is at (−4, 0) .
D. The axis of symmetry of h(t) is t=−3/2
1. The width that will maximize the area of the rectangle can be found by finding the x-coordinate of the vertex of the quadratic function. In this case, the vertex can be found at w = 8 meters. Therefore, the answer is A. 8 meters.
2. The horizontal length of the jump can be determined by finding the x-intercepts of the quadratic function. In this case, the x-intercepts can be found at d = -2 and d = 2. Therefore, the horizontal length of the jump is 4 meters. The maximum height can be found by finding the y-coordinate of the vertex of the quadratic function. In this case, the vertex can be found at (0, 1). Therefore, the maximum height is 1 meter. Therefore, the answer is D. The horizontal length of the jump is 4 meters and the maximum height is 1 meter.
3. By graphing the function f(x) = 1/2x^2 - 2x using technology, we can see that the vertex is at (2, -2) and the y-intercept is at (0, 0). Therefore, the statement "The vertex and the y-intercept of f(x) are at the same point" is false. Therefore, the answer is not A.
4. By graphing the function h(t) = -2t^2 + 6t - 4 using technology, we can see that the vertex is at (1.5, 0.5), the zeros are at (-1, 0) and (2, 0), and the y-intercept is at (0, -4). Therefore, the statement "The vertex of h(t) is at (3/2, 1/2)" is false. Therefore, the answer is not A.
1. To find the width that maximizes the area of the rectangle, we need to find the vertex of the quadratic function A(w)=w(16-w). The vertex of a quadratic function in the form of f(x) = ax^2 + bx + c is given by the formula (-b/2a, f(-b/2a)). In this case, a = 1, b = -16, and c = 0.
Applying the formula, we get:
w = -(-16)/2(1) = 16/2 = 8 meters
Therefore, the width that maximizes the area of the rectangle is 8 meters.
Answer: A. 8 meters
2. To find the horizontal length and maximum height of the jump, we need to analyze the given function h(d) = -1/4d^2 + d.
The horizontal length is determined by finding the zeros of the function h(d), where h(d) = 0. To find the zeros, we can set h(d) = -1/4d^2 + d = 0 and solve for d.
-1/4d^2 + d = 0
d(-1/4d + 1) = 0
Either d = 0 or -1/4d + 1 = 0.
Solving -1/4d + 1 = 0, we get:
-1/4d = -1
d = 4
Therefore, the horizontal length of the jump is 4 meters.
To find the maximum height, we need to find the y-coordinate of the vertex of the function h(d) = -1/4d^2 + d. The vertex of a quadratic function in the form of f(x) = ax^2 + bx + c is given by the formula (-b/2a, f(-b/2a)). In this case, a = -1/4 and b = 1.
Applying the formula, we get:
d = -1/((2)(-1/4)) = -1/(-1/2) = -2
Substituting -2 into the function h(d), we get:
h(-2) = -1/4(-2)^2 + (-2)
= -1/4(4) - 2
= -1 - 2
= -3
Therefore, the maximum height of the jump is 3 meters.
Answer: C. The horizontal length of the jump is 4 meters and the maximum height is 1/4 meter.
3. Using technology to graph the function f(x)=1/2x^2 - 2x, we can see that the vertex and the y-intercept of f(x) are at different points. The vertex is the lowest point on the graph, while the y-intercept is a point where the graph intersects the y-axis.
Therefore, the true statement is:
Answer: A. The vertex and the y-intercept of f(x) are at the same point.
4. The y-intercept is the point where the graph of h(t) intersects the y-axis. We can see from the graph that the y-intercept is at (0, -4).
The zeros of h(t) are the points where the graph intersects the x-axis. We can see from the graph that the zeros are at (-1, 0) and (2, 0).
Therefore, the true statement is:
Answer: C. The y-intercept of h(t) is at (-4, 0).
1. To find the width that maximizes the area of the rectangle, we need to determine the vertex of the quadratic function. The vertex of a quadratic function in the form of A(w) = aw^2 + bw + c can be found using the formula w = -b / (2a). In this case, a = 16 and b = -16.
Substituting these values into the formula, we get:
w = -(-16) / (2 * 16)
w = 16 / 32
w = 0.5 meters
However, since the width represents the physical dimension of the rectangle, we cannot have a width of 0.5 meters. Therefore, we can conclude that the width so that the area of the rectangle is at its maximum is 0.5 meters or option A is incorrect.
2. To determine the horizontal length and maximum height of the jump, we need to find the vertex of the quadratic function h(d) = -1/4d^2 + d. The vertex of a quadratic function in the form of h(d) = ad^2 + bd + c can be found using the formula d = -b / (2a). In this case, a = -1/4 and b = 1.
Substituting these values into the formula, we get:
d = -(1) / (2 * -1/4)
d = -4 / -1/2
d = -4 * -2
d = 8 meters
So, the horizontal length of the jump is 8 meters. In order to find the maximum height, we substitute the value of d = 8 into the function h(d):
h(8) = -1/4(8)^2 + (8)
h(8) = -1/4 * 64 + 8
h(8) = -16 + 8
h(8) = -8
Therefore, the maximum height of the jump is -8 meters. However, since the height cannot be negative, we conclude that the maximum height is 8 meters or option B is incorrect.
3. To graph the function f(x) = 1/2x^2 - 2x, you can use mathematical software or graphing calculators like Desmos or GeoGebra. Plotting the function, you will see that the graph is a downward opening parabola. The vertex of the parabola represents the maximum or minimum point of the function.
Now, to determine the true statement, we need to examine the graph. In this case, option A is correct. The vertex and the y-intercept of f(x) are at the same point.
4. The given quadratic function h(t) = -2t^2 + 6t - 4 represents the height of the dolphin relative to the water level at time t seconds. To answer the question, we need to analyze the graph. From the graph, we can make the following observations:
- The vertex of the parabola represents the maximum or minimum point of the function.
- The zeros of the function represent the points where the graph intersects the x-axis.
- The y-intercept is the point where the graph intersects the y-axis.
From the graph of h(t), we can determine the following:
- The vertex of h(t) is at the point (3/2, 1/2), which means option A is incorrect.
- The zeros of h(t) are at approximately (-1, 0) and (2, 0), so option B is incorrect.
- The y-intercept of h(t) is at the point (0, -4), so option C is incorrect.
- The axis of symmetry of h(t) is t = 3/2, so option D is correct.
Therefore, the correct statement is that the axis of symmetry of h(t) is t = 3/2.