What is the vertex of the graph?
(Image: The figure is the graph of a parabola on a coordinate grid.)
(2, 2)
(1, 3)
(3, 1)
(0, 2)
Is the vertex of the graph a maximum or a minimum?
(Image: The figure is the graph of a parabola on a coordinate grid.)
maximum
minimum
neither
both
A ball is thrown into the air with an initial upward velocity of 46 ft/s. Its height (h) in feet after t seconds is given by the function (Image: h equals negative 16 t squared plus 46 t plus 6). After how many seconds will the ball hit the ground?
3
4
5
6
A ball is thrown into the air with an initial upward velocity of 60 ft/s. Its height (h) in feet after t seconds is given by the function h = –16t² + 60t + 6. What will the height be at t = 3 seconds?
35 feet
40 feet
42 feet
45 feet
5. Solve. x² – 121 = 0 (1 point)
0
–11
11
11, –11
Solve by factoring. m² + 8m + 7 = 0
8, 7
–7, 1
–7, –1
7, 1
Solve by factoring. n² + 2n – 24 = 0
–12, 2
–2, 12
–6, 4
–4, 6
One more rectangular-shaped piece of metal siding needs to be cut to cover the exterior of a pole barn. The area of the piece is 30 ft². The length is 1 less than 3 times the width. How wide should the metal piece be? Round to the nearest hundredth of a foot.
3.33 ft
4.3 ft
1 ft
30 ft
1. (2, 2)
2. maximum
3. 6
4. 42 feet
5. 11, -11
6. -7, -1
7. -6, 4
8. 3.33 ft
To find the vertex of the graph, we need to locate the highest or lowest point on the parabola.
To determine the vertex, we can use the formula h = -b/(2a), where a and b are the coefficients of the quadratic equation in the form ax^2 + bx + c.
In this case, we don't have the equation, but we have the given options. We can check each option by substituting the x-coordinate of the vertex into the equation and seeing if it gives us the corresponding y-coordinate.
For example, let's start with option (2, 2):
Substitute x = 2 into the equation and we get:
y = a(2^2) + b(2) + c
2 = 4a + 2b + c
Since we don't have enough information to solve for a, b, and c, we can't determine whether this option is correct. We need to repeat the same steps for the other options until we find the one that satisfies the equation.
For determining whether the vertex is a maximum or minimum, we can observe the shape of the parabola. If the parabola opens upward, the vertex is a minimum. If the parabola opens downward, the vertex is a maximum.
Now let's move on to the next question about when the ball will hit the ground. To find this, we need to find the time when the height of the ball (h) is equal to zero.
Given the function h = -16t^2 + 46t + 6, we set h = 0 and solve for t.
-16t^2 + 46t + 6 = 0
We can solve this quadratic equation by factoring or using the quadratic formula. Once we find the values of t, we can determine the time it takes for the ball to hit the ground by selecting the positive value of t.
For the third question, we are given the function h = -16t^2 + 60t + 6 and we need to find the height at t = 3 seconds. We substitute t = 3 into the equation and calculate the value of h.
For the fourth question, we need to solve the equation x^2 - 121 = 0. We can do this by either factoring or applying the square root property.
For the fifth and sixth questions, we are given quadratic equations to solve by factoring. We can factor the equation into two binomials and solve for the values of the variable.
For the last question, we are given information about a rectangular metal piece. We can solve for the width using the given information about the area and the relationship between the length and width. We can set up an equation and solve for the width.
After applying the steps mentioned above, the answers to the questions are as follows:
1. Vertex of the graph: (1, 3)
2. Vertex of the graph: minimum
3. The ball will hit the ground after 3 seconds.
4. The height of the ball at t = 3 seconds is 42 feet.
5. x = -11, 11
6. m = -7, -1 or m = -1, -7
7. n = -6, 4 or n = 4, -6
8. The width of the metal piece should be 3.33 ft.