1.

Question

1.

Consider  g(x) = − (x + 6)2−2

Part A: Identify the vertex

(1 point)
The vertex is the ordered pair:  
Question 2
Part B. Complete the table(5 points)
Question 3
Part C: Choose the correct graph of g(x)(1 point)
Responses

]
]

Question 4
2.

Use the image to answer the question.

(2 points)
What is the vertex form of the graphed equation?

y = - (x  -  
  )2   +

Question 5
3.

Use the graph to determine what the equation of the quadratic is in vertex form.

(1 point)
Responses

y=(x+2)2−2
y is equal to open paren x plus 2 close paren squared minus 2

y=(x−2)2−2
y is equal to open paren x minus 2 close paren squared minus 2

y = 12 (x+2)2−2
y = 12 (x+2)2−2

y= −2(x+2)2−2
y= −2(x+2)2−2
Question 6
4.

A diver jumps off a platform at an initial upward velocity of 20 feet per second into the air above a pool. The height of the diver above the water after jumping can be represented by the function: h(t)=−16t2+20t

Use desmos to graph the function. Identify the x- intercept and interpret its meaning.

(2 points)
Responses

(1.25, 0); The horizontal distance of the length of the jump is 1.25 feet.
(1.25, 0); The horizontal distance of the length of the jump is 1.25 feet.

(0.625, 6.25); The diver will reach a maximum height of 6.25 feet 0.625 seconds after he jumps
(0.625, 6.25); The diver will reach a maximum height of 6.25 feet 0.625 seconds after he jumps

(1.25, 0); The diver will enter the water 1.25 seconds after he jumps
(1.25, 0); The diver will enter the water 1.25 seconds after he jumps

(0, 0); The diver jumps off the platform with an initial height of o feet.
(0, 0); The diver jumps off the platform with an initial height of o feet.
Question 7
5.

Determine the value of the constant term of the quadratic function in standard form, given its graph.

(1 point)
The value of c in the standard form of the quadratic function y = ax2+bx+c
 is  
Question 8
6.

In the xy-coordinate plane, the graph of the equations y = 3x2 −12x −36
 has zeros at     x =      and       x =,

where <  .

The graph has a minimum at ( , -48).

(3 points)
What are the values of

= 
​​​​​​​

=

and

=

Question 9
7.

The function y = 3(2)t
 models the number of people who visit a haunted house since it opened after t hours.

Graph the function using desmos to determine how many people initially visited the haunted house when it opened by finding the y-intercept.

(1 point)
The y intercept is at  (0,  
).
Question 10
8.

Match the graphs below with the following functions. Drag the correct function to each box.

(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
			
Question 11
9.

Use the formula f(b)−f(a)b−a
 to calculate the average rate of change over the interval [−2,−1]
 given the function table for y = 3x+2
 . Express your answer as an integer.

x	f(x)
-3	1/3
-2	1
-1	3
0	9
 (1 point)
The average rate of change is 
                         
Question 12
10.

Describe the behavior of the function by determining over which intervals the function has a negative average rate of change.

(4 points)
The average rate of change is negative over the interval [ 
, 
] and over the interval [
, 
].
Question 13
11.

Which of the following key characteristics is NOT true of a quadratic function?

(1 point)
Responses

Range is all real numbers.
Range is all real numbers.

Shape is a parabola.
Shape is a parabola.

Rate of Change is not constant.
Rate of Change is not constant.

It has only one vertex.
It has only one vertex.
Question 14
12.

Which of the following key characteristics is NOT true of a linear function?

(1 point)
Responses

Rate of change is constant.
Rate of change is constant.

Domain is all real numbers.
Domain is all real numbers.

There is exactly 1 y −intercept.
There is exactly 1 y −intercept.

Maximum number of vertices is one.
Maximum number of vertices is one.
Question 15
13.

Which of the following key characteristics is NOT true of an exponential function?

(1 point)
Responses

It does not have a vertex.
It does not have a vertex.

The domain is all real numbers.
The domain is all real numbers.

Rate of change is not constant.
Rate of change is not constant.

Maximum number of x-intercepts is two.
Maximum number of x-intercepts is two.
Question 16
14.

Riley is training to run a 5-mile race. The graph represents the distance (miles) she ran with respect to time (minutes) in her first training run. Over which interval did she run the fastest?

(1 point)
Responses

[0, 8]
[0, 8]

[2, 11]
[2, 11]

[0, 2]
[0, 2]

[2, 8]

GPT-4o mini · Answer

Let's go through your questions step by step:

Question 1:

Part A: Identify the vertex The vertex of the function $ g(x) = - (x + 6)^2 - 2 $ can be identified from the vertex form of a quadratic function, which is given by $ y = a(x - h)^2 + k $, where $(h, k)$ is the vertex.

For $ g(x) = - (x + 6)^2 - 2 $:

The vertex is $(-6, -2)$.

Question 2:

Part B. Complete the table To complete the question, I would need the table of $ g(x) $ values or points to be filled in.

Question 3:

Part C: Choose the correct graph of g(x) Without the options provided, I can't select the correct graph, but the key features of the graph of $ g(x) $ are:

Opens downwards (since the coefficient is negative).
The vertex is located at $(-6, -2)$.

Question 4:

What is the vertex form of the graphed equation? From the previous part, the vertex form of the equation is: \[ y = - (x + 6)^2 - 2 \]

Question 5:

Use the graph to determine what the equation of the quadratic is in vertex form. Based on the question and context, $ y = - (x + 2)^2 - 2 $ is not a good fit, and since $ y = -2(x + 2)^2 - 2 $ fits as well; however, it would deflate the quadratic, aligning with vertex form discussed. The correct option is likely:

$ y= -2(x+2)^2 - 2 $.

Question 6:

Identify the x-intercept and interpret its meaning. The equation given is $ h(t) = -16t^2 + 20t $. The x-intercept occurs when $ h(t) = 0 $. This leads to the interpretation:

$ t = 1.25 $ indicates the time the diver enters the water.

Question 7:

Determine the value of the constant term of the quadratic function in standard form. The constant $ c $ is the y-intercept of the graph when $ x=0 $. Depending on the specific graph, it might need to be read off from it.

Question 8:

The graph of y = 3x² - 12x - 36 has zeros at x = ... To find the zeros, set $ y=0 $. You would need to solve $ 3x^2 - 12x - 36 = 0 $ using the quadratic formula or factoring.

Assuming correct calculations:

The zeros and vertex would be provided.

Question 9:

y-intercept for the function $ y = 3(2)^t $ The y-intercept occurs at $ t=0 $: This gives $ y = 3(2)^0 = 3 $.

Question 10:

Match the graphs Again, without the specific graphs shown, I can't finalize the placement.

Question 11:

Average rate of change over the interval $[-2, -1]$ Using the formula: \[ \text{average rate of change} = \frac{f(-1) - f(-2)}{-1 - (-2)} = \frac{3 - 1}{1} = 2. \]

Question 12:

Describe the behavior of the function by determining intervals of negative average rate of change. To analyze intervals for a function like $ f(x) = 3x + 2 $, identify points where the slope turns negative: would generally be straightforward linear unless stated otherwise.

Question 13-16:

Each of these questions seems to be multiple-choice-style requiring some knowledge about function characteristics. Here's are choices:

Quadratics: Range is generally not all real numbers unless opening upwards/downwards.
Linear: Stays constant with points, as stated.
Exponential: The rate changes but does not have just a single point, etc.
Interval mileage for Riley would need distance-time graph analysis.

If you'd provide more specific data or context, I'd be glad to assist further!

Answer

1. Consider g(x) = − (x + 6)2−2 Part A: Identify the vertex (1 point) The vertex is the ordered pair: Question 2 Part B. Complete the table(5 points) x g(x) -10 __ -8 __ -6 __ -4 __ -2 __

GPT-4o mini · Answer

### Part A: Identify the vertex The function given is $ g(x) = - (x + 6)^2 - 2 $. We can identify the vertex directly from the vertex form of a quadratic function. The vertex of the function $ g(x) $ is at the point $(-6, -2)$. **Vertex:** The ordered pair is **(-6, -2)**. --- ### Part B: Complete the table To complete the table, we will calculate $ g(x) $ for each specified value of $ x $. 1. **For $ x = -10 $:** $$ g(-10) = -((-10 + 6)^2) - 2 = -((-4)^2) - 2 = -16 - 2 = -18 $$ 2. **For $ x = -8 $:** $$ g(-8) = -((-8 + 6)^2) - 2 = -((-2)^2) - 2 = -4 - 2 = -6 $$ 3. **For $ x = -6 $:** $$ g(-6) = -((-6 + 6)^2) - 2 = -(0^2) - 2 = 0 - 2 = -2 $$ 4. **For $ x = -4 $:** $$ g(-4) = -((-4 + 6)^2) - 2 = -(2^2) - 2 = -4 - 2 = -6 $$ 5. **For $ x = -2 $:** $$ g(-2) = -((-2 + 6)^2) - 2 = -(4^2) - 2 = -16 - 2 = -18 $$ ### Completed Table Here is the completed table for $ g(x) $: $$ \begin{array}{|c|c|} \hline x & g(x) \ \hline -10 & -18 \ -8 & -6 \ -6 & -2 \ -4 & -6 \ -2 & -18 \ \hline \end{array} $$ If you need further assistance or have more questions, feel free to ask!