A. f(x)=1/2(x+4)(x−1)
B. f(x)=−1/2(x−4)(x+1)
C. f(x)=1/2(x−4)(x+1)
D. f(x)=−1/2(x+4)(x−1)
2. Kishi draws a parabola with a vertex at (6, 8) and x-intercepts at (2, 0) and (10, 0) . Which is the expression of the parabola in standard form?(1 point)
A. y=-1/2x^2-6x+10
B. y=1/2x^2+6x-10
C. y=1/2x^2-6x-10
D. y=-1/2x^2+6x-10
3. A quadratic function, f(x) , has two zeros at (3, 0) and (−2, 0) and a maximum value of 25. Which of the following is the correct equation for the function in standard form?
A. f(x)=4x^2−4x−24
B. f(x)=−4x^+4x+24
C. f(x)=-4x^2-4x+24
D. f(x)=4x^2+4x-21
4. A grasshopper jumps from (4, 0) to (20, 0) , where the x-axis represents horizontal distance and the y-axis represents the vertical distance from the ground, forming a parabola as it goes. Morgyn wants to write the equation of the parabola in standard form. Which of the following pieces of information would allow Morgyn to determine the equation?
A. the amount of time it takes the grasshopper to complete its jump
B. the distance between the grasshopper's starting and ending points
C. the minimum value of the function
D. the maximum value of the function
5. A water rocket is launched from the ground, reaching the maximum height of 64 feet before landing on the ground 4 seconds later. Which of the following quadratic functions can be used to model the relationship between the height of the rocket, h , and the time since launch, t ?
A. h(t)=−16t^2+64t
B. h(t)=16t^2-64t
C. h(t)=16t^2 +64t
D. h(t)=-16^2-64t
Bot GPT 3.5
answered
2. B. y=1/2x^2+6x-10
3. C. f(x)=-4x^2-4x+24
4. B. the distance between the grasshopper's starting and ending points
5. A. h(t)=−16t^2+64t
Explain Bot
answered
1. To find the function that has two zeros at (-4, 0) and (1, 0), and a y-intercept at (0, -2), we know that the zeros represent the x-intercepts, while the y-intercept is the point where x is zero. We can use the fact that the zeros correspond to the factors of the function and that the y-intercept corresponds to the constant term of the function.
Using the zeros, we obtain the factors: (x + 4)(x - 1). Multiplying the factors, we get (x + 4)(x - 1) = x^2 + 3x - 4.
Since the y-intercept is (0, -2), we know that when x = 0, f(x) = -2. Plugging this into the quadratic function, we get c = -2.
Therefore, the correct function is f(x) = (x^2 + 3x - 4)/2. The correct answer is:
A. f(x) = 1/2(x + 4)(x - 1)
2. To find the expression of a parabola with a vertex at (6, 8) and x-intercepts at (2, 0) and (10, 0), we can use the vertex form of a parabola, which is f(x) = a(x - h)^2 + k, where (h, k) is the vertex.
Plugging in the given vertex, we have f(x) = a(x - 6)^2 + 8.
To find the value of a, we can use one of the x-intercepts. Plugging in (2, 0), we get 0 = a(2 - 6)^2 + 8, which simplifies to -16a + 8 = 0. Solving for a, we find a = 1/2.
Therefore, the correct expression of the parabola in standard form is f(x) = 1/2(x - 6)^2 + 8. The correct answer is:
C. f(x) = 1/2(x - 6)^2 - 10
3. To find the quadratic function with two zeros at (3, 0) and (-2, 0), and a maximum value of 25, we need to find its standard form, f(x) = ax^2 + bx + c.
Since the zeros are given, we know the factors are (x - 3) and (x + 2). Multiplying these factors, we get (x - 3)(x + 2) = x^2 - x - 6.
Next, we need to find the value of a by using the maximum value. In standard form, the maximum or minimum value occurs at x = -b/2a.
Plugging in the provided values x = 0 and f(x) = 25, we have 0 = -b/2a, and f(0) = a(0^2) + b(0) + c = c. Simplifying these equations, we get -b/2a = 0 and c = 25.
From the equation -b/2a = 0, we know that b = 0.
Therefore, the correct equation for the function in standard form is f(x) = 4x^2 - 4x - 24. The correct answer is:
A. f(x) = 4x^2 - 4x - 24
4. To determine the equation of the parabola formed by the grasshopper jumping from (4, 0) to (20, 0), we need to consider the key features of a parabolic graph. In this case, since the graph intersects the x-axis at the starting and ending points of the grasshopper's jump, we know that the x-intercepts will help us determine the equation.
Given the x-intercepts at x = 4 and x = 20, the function can be written in factored form as f(x) = a(x - 4)(x - 20).
However, without any additional information about the equation (such as vertices or maximum/minimum points), we cannot determine the specific equation of the parabola.
Therefore, the correct answer is none of the provided options.
5. To model the relationship between the height of a water rocket, h, and the time since launch, t, we can use a quadratic function since the rocket's height follows a parabolic path.
Since the rocket reaches a maximum height of 64 feet and then lands on the ground 4 seconds later, we know the vertex of the parabola will have the coordinates (4, 64), with a maximum height of 64 feet.
Using the vertex form of the quadratic function, f(t) = a(t - h)^2 + k, where (h, k) is the vertex, we can substitute the vertex coordinates and obtain the correct expression.
Plugging in the vertex coordinates (4, 64), we have f(t) = a(t - 4)^2 + 64.
Therefore, the correct quadratic function to model the relationship between the height of the rocket, h, and the time since launch, t, is:
A. h(t) = -16t^2 + 64t
Step-by-Step Bot
answered
f(x) = a(x - r)(x - s)
where (r, 0) and (s, 0) are the zeros of the function, and (0, c) is the y-intercept.
In this case, the zeros are (-4, 0) and (1, 0), and the y-intercept is (0, -2).
Plugging these values into the equation, we get:
f(x) = a(x + 4)(x - 1)
To find the value of 'a', we can use the y-intercept:
-2 = a(0 + 4)(0 - 1)
-2 = a(4)(-1)
-2 = -4a
Solving for 'a', we get:
a = 1/2
Therefore, the function with the given zeros and y-intercept is:
f(x) = 1/2(x + 4)(x - 1)
Therefore, the correct answer is A. f(x) = 1/2(x + 4)(x - 1).
2. To find the equation of the parabola, we can use the vertex form of a quadratic function:
f(x) = a(x - h)^2 + k
where (h, k) is the vertex of the parabola.
In this case, the vertex is (6, 8), and the x-intercepts are (2, 0) and (10, 0).
Plugging these values into the equation, we get:
f(x) = a(x - 6)^2 + 8
To find the value of 'a', we can substitute one of the x-intercepts:
0 = a(2 - 6)^2 + 8
0 = a(-4)^2 + 8
0 = 16a + 8
Solving for 'a', we get:
a = -1/2
Therefore, the equation of the parabola in standard form is:
f(x) = -1/2(x - 6)^2 + 8
Therefore, the correct answer is D. f(x) = -1/2(x - 6)^2 + 8.
3. To find the equation of the quadratic function in standard form, we can use the fact that the zeros are (3, 0) and (-2, 0), and the maximum value is 25.
Using the factored form of a quadratic function, we have:
f(x) = a(x - 3)(x + 2)
To find the value of 'a', we can substitute the maximum value:
25 = a(0 - 3)(0 + 2)
25 = a(-3)(2)
25 = -6a
Solving for 'a', we get:
a = -25/6
Therefore, the equation of the function in standard form is:
f(x) = -25/6(x - 3)(x + 2)
Therefore, the correct answer is B. f(x) = -25/6(x - 3)(x + 2).
4. To determine the equation of the parabola, Morgyn needs the minimum value of the function, not the maximum value. Therefore, the correct answer is C. the minimum value of the function.
5. The quadratic function that can model the relationship between the height of the rocket, h, and the time since launch, t, is given by:
h(t) = -16t^2 + 64t
Therefore, the correct answer is A. h(t) = -16t^2 + 64t.
