An owl swoops down from its perch in a tree at 20 feet to catch field mice below. It follows the function h=−6t2−24t+20

Question

. Plot points to graph the equation. Which ordered pair represents the vertex?(1 point)
Responses

(−5,0)
left parenthesis negative 5 comma 0 right parenthesis

(−2,44)
left parenthesis negative 2 comma 44 right parenthesis

(1,0)
left parenthesis 1 comma 0 right parenthesis

(44,−2)

Answer 1

To find the vertex of the quadratic function given by \( h(t) = -6t^2 - 24t + 20 \), we can either complete the square or use the formula for the vertex of a parabola. The vertex \( (h, k) \) of a quadratic function in the standard form \( ax^2 + bx + c \) can be found using the formulas:

\[ t = -\frac{b}{2a} \] \[ h = f(t) \]

In this case, \( a = -6 \), \( b = -24 \), and \( c = 20 \).

1. Calculate \( t \): \[ t = -\frac{-24}{2 \cdot -6} = \frac{24}{-12} = -2 \]

2. Calculate \( h \) at \( t = -2 \): \[ h(-2) = -6(-2)^2 - 24(-2) + 20 \] \[ = -6(4) + 48 + 20 \] \[ = -24 + 48 + 20 \] \[ = 24 \]

So the vertex is at \( (-2, 24) \).

Therefore, none of the options in your list seem to represent the vertex correctly, but if we look closely, it seems like there might be a mistake in the available options. If "left parenthesis negative 2 comma 44 right parenthesis" is intended as \( (-2, 44) \), then that would be incorrect, as we found the vertex to be \( (-2, 24) \).

Thus, based on the calculations, the ordered pair representing the vertex is \( (-2, 24) \), which is not listed as one of the options you've provided. If you're limited to those choices, the calculations lead to \( (-2, 24) \) nonetheless.

Answer 2

Luca is building a garden in his backyard. He needs to erect a fence to keep his dog Luna from digging in it. He has enough materials for a fence with 120 feet around the perimeter. This function is represented by the formula A=−w2+60w
, where A
is the area and w is the width. Which of the following correctly identifies the x-intercept(s)?(1 point)
Responses

(0,0)
and (60,0)
left parenthesis 0 comma 0 right parenthesis and left parenthesis 60 comma 0 right parenthesis

(60,0)
and (30,900)
left parenthesis 60 comma 0 right parenthesis and left parenthesis 30 comma 900 right parenthesis

(0,0)
left parenthesis 0 comma 0 right parenthesis

(60,0)

Answer 3

To find the x-intercepts of the function \( A = -w^2 + 60w \), we set \( A \) to 0 and solve for \( w \):

\[ 0 = -w^2 + 60w \]

Rearranging this gives:

\[ w^2 - 60w = 0 \]

Now we can factor this equation:

\[ w(w - 60) = 0 \]

Setting each factor equal to 0, we find:

1. \( w = 0 \)
2. \( w - 60 = 0 \) which gives \( w = 60 \)

Thus, the x-intercepts are \( (0, 0) \) and \( (60, 0) \).

From the options provided, the correct answer is:

(0,0) and (60,0)

This means that Luca's fence can have an area of 0 when the width is either 0 feet (meaning there is no garden) or when the width reaches its maximum of 60 feet.

Answer 4

An owl swoops down from its perch in a tree at 30 feet to catch field mice. It follows the function h=6t2 −24t+30
, where t is the time in seconds and h is the height in feet. Which of the following best interprets the vertex?(1 point)
Responses

(0,30)
left parenthesis 0 comma 30 right parenthesis

(30,0)
left parenthesis 30 comma 0 right parenthesis

(6,2)
left parenthesis 6 comma 2 right parenthesis

(2,6)

Answer 5

To find the vertex of the quadratic function \( h(t) = 6t^2 - 24t + 30 \), we can use the vertex formula:

\[ t = -\frac{b}{2a} \]

where \( a = 6 \) and \( b = -24 \).

1. Calculate \( t \): \[ t = -\frac{-24}{2 \cdot 6} = \frac{24}{12} = 2 \]

2. Calculate \( h \) at \( t = 2 \): \[ h(2) = 6(2)^2 - 24(2) + 30 \] \[ = 6(4) - 48 + 30 \] \[ = 24 - 48 + 30 \] \[ = 6 \]

So, the vertex of the function is at the point \( (2, 6) \).

Interpretation of the Vertex

In the context of the problem:

• The \( t \)-coordinate of the vertex (2) represents the time in seconds at which the owl reaches its maximum height.
• The \( h \)-coordinate of the vertex (6) represents the maximum height of the owl during its swoop, which is 6 feet.

Therefore, the vertex \( (2, 6) \) indicates that 2 seconds after the owl starts swooping, it reaches a height of 6 feet (before it descends further).

From the given options, the best interpretation is:

(2, 6)