1.

Question

1.

To show how to solve the equation by graphing, create two functions and find their intersection points. What two functions can be used to solve the following equation, and what is their solution set?

10−x2=x+4

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(1 point)
Responses

g(x) = x + 4, f(x) = 10+x2, {−3,2}
g(x) = x + 4, f(x) = 10+x2, {−3,2}

g(x) = x + 4, f(x) = 10+x2, {−3,1}
g(x) = x + 4, f(x) = 10+x2, {−3,1}

g(x) = x + 4, f(x) = 10−x2, {1,6}
g(x) = x + 4, f(x) = 10−x2, {1,6}

g(x) = x + 4, f(x) = 10−x2, {−3,2}
g(x) = x + 4, f(x) = 10−x2, {−3,2}
Question 2
2.

What function should be graphed in order to solve the equation 9x−6  =−8
?

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(1 point)
Responses

f(x) = 9x + 2
f(x) = 9x + 2

f(x) = 9x - 14
f(x) = 9x - 14

f(x) =17x - 6
f(x) =17x - 6

f(x) =x -6
f(x) =x -6
Question 3
3.

What is the solution to the equation 2x - 1 = 4x + 3?

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(1 point)
x = 
Question 4
4.

You are traveling home from work. You are decreasing the distance as you walk home. Your house is 41 blocks away, and you walk 3 blocks per minute.

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(4 points)
Your rate of change in this situation is 
 (only answer with an integer)

Your starting amount is 
 (Only answer with an integer)

The equation that represents the situation in slope intercept form is y=

It will take 
 minutes (to the nearest minute), to get home.

Question 5
5.

A water tank is being emptied and replaced with another one.

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(7 points)
How much water did the tank have when it started being drained? 
 This would represent the 
 of the line graphed.

The tank is empty after 
 . This would be the 
 of the line graphed.

The tank is emptying at a rate of 
 . This represents a(n) 
 slope

The equation that would represent this scenario would be

Question 6
6.

Brandi and her daughter, Ella, are training for a hiking challenge. Because Brandi hikes at a slower pace than her daughter, she begins the practice hike two hours earlier. If Brandi averages a pace of 4 mph, the linear equation y=4x can be used to model her distance, y, in miles with respect to her time, x, in hours. If Ella averages a pace of 6 mph and begins two hours after her mom, the linear equation y=6x-12 can be used to model her distance, y, in miles with respect to time, x, in hours. The graph of which two lines can be used to find the time and distance when Ella catches up with her mother?

Desmos Link

(1 point)
Responses

Line 1 and Line 2
Line 1 and Line 2

Line 2 and Line 3
Line 2 and Line 3

Line 3 and Line 4
Line 3 and Line 4

Line 1 and Line 4
Line 1 and Line 4
Question 7
7.

A radioactive substance decays at a rate of 6% each year. If the initial amount of the substance was 600 grams, create an exponential function to model the decay of the substance. Which of the following correctly models the decay?

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(1 point)
Responses

f(x) = 600(0.06)x
f(x) = 600(0.06)x

f(x) = 600(.40)x
f(x) = 600(.40)x

f(x) = 600(0.6)x
f(x) = 600(0.6)x

f(x) = 600(0.94)x
f(x) = 600(0.94)x
Question 8
8.

Two college roommates are studying an invasive species at a local park. What was the size of the population when they began recording data on the invasive species?

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(1 point)
The starting amount was  
  species. Round to the nearest tens.
Question 9
9.

Lorraine hits a golf ball into the air. It lands 60 feet away from her. At its highest point, the ball is 30 feet in the air.

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(6 points)
This scenario would best be represented by a 
 equation.

The ball lands 60 feet away from her would represent the 
. Half of this distance would be axis of symmetry. That would make the vertex

The ball starts at how many feet away from Lorraine? 
  So another point on the graph would be

Using the information, the only value that is not given and needs to be found is the

Question 10
What is the equation for this scenario?

What is the height of the golf ball when it is 15 feet away from Lorrain?

(2 points)
The equation is f(x) = 
 
 (answer using fractions and do not include grouping symbols)

The height of the ball when it is 15 feet away from Lorraine is 
 feet in the air. (round to the nearest tenth)

Question 11
10.

While hiking, Julia kicked a small rock off a ledge that was meters above a crevasse. Graph the equation f(x) = −4.9x2+9
. Use the graph to determine when the rock will hit the bottom of the crevasse, 2 meters below ground level. Round your answer to the nearest tenth of a second.

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(1 point)
The rock will hit the bottom of the crevasse in 
 seconds.
Question 12
11.

Function 1: f(x) = 32x + 5

Function 2:

Desmos Link

(1 point)
Function 
 has a greater rate of change.

GPT-4o mini · Answer

Let's address each question one by one:

Question 1

To solve the equation \(10 - x^2 = x + 4\) by graphing, we first need to rewrite it in the form of two functions:

\(f(x) = 10 - x^2\) (this is a downward opening parabola)
\(g(x) = x + 4\) (this is a linear function)

To find the solution, we look for the intersection of these two functions. Setting the equations equal: \[ 10 - x^2 = x + 4. \] Rearranging gives us: \[ x^2 + x - 6 = 0. \] Factoring gives: \[ (x - 2)(x + 3) = 0. \] Thus, the solutions are: \[ x = 2 \quad \text{and} \quad x = -3. \] Answer: \(g(x) = x + 4, f(x) = 10 - x^2, {−3,2}\)

Question 2

To solve \(9x - 6 = -8\), we can rearrange the equation: \[ 9x - 6 + 6 = -8 + 6 \implies 9x = -2. \] Hence, the function to graph is: \[ f(x) = 9x + 2, \] to find the x-value when this equals zero.

Answer: \(f(x) = 9x + 2\)

Question 3

To solve the equation \(2x - 1 = 4x + 3\), rearranging gives: \[ 2x - 4x = 3 + 1 \implies -2x = 4 \implies x = -2. \]

Answer: \(x = -2\)

Question 4

Your rate of change: 3 (blocks per minute)
Your starting amount: 41 (blocks)
Equation: \(y = -3x + 41\) (where \(y\) represents distance remaining)
To find how long it takes to walk home: \(0 = -3x + 41\): \[ 3x = 41 \implies x = \frac{41}{3} \approx 14 \text{ minutes}. \]

Answer:

Rate of change: 3
Starting amount: 41
Equation: \(y = -3x + 41\)
Minutes: 14

Question 5

Assuming the tank starts with an initial amount \(A\), at a steady rate of decrease \(r\), then it can be said:

Starting amount: \(A\)
Empty after: time \(t\)
Rate of emptying: \(r\) (you have to provide the numeric values)
Equation: \(y = A - rt\)

(Specific values would need to be provided to fill in the blanks.)

Question 6

To find the time and distance when Ella catches up, we look at her equation \(y = 6x - 12\) and Brandi's \(y = 4x\). Therefore:

Answer: Line 1 and Line 2

Question 7

The decay function of a substance that decreases at a rate of 6% is modeled as: \[ f(x) = 600(1 - 0.06)^x = 600(0.94)^x. \]

Answer: \(f(x) = 600(0.94)^x\)

Question 8

For this question, without the specific data provided in your reference, I cannot specify the starting population. Please provide the number or additional data.

Question 9

This scenario represents a parabolic equation because it describes the motion of the ball.

The maximum height of 30 feet is the vertex point.
The distance of 60 feet is the x-intercept.
Axis of symmetry: \(x = 30\).
The point where the ball starts is \(x = 0\).

Answer: You will need to determine the exact equation using the vertex form or standard form based on data provided.

Question 10

I cannot answer this without additional specifics about the structure of the equation you wish to use for the scenario; please provide more context.

Question 11

To find when \(f(x) = -4.9x^2 + 9\) hits 2 meters, you would solve: \[ -4.9x^2 + 9 = -2 \implies -4.9x^2 + 9 + 2 = 0. \] Rearranging gives: \[ -4.9x^2 + 11 = 0 \implies 4.9x^2 = 11 \implies x^2 = \frac{11}{4.9} \implies x = \sqrt{\frac{11}{4.9}}. \] Calculating this gives the time when the rock hits the bottom.

Question 12

Without specific details on Function 2, I can't determine which function has a greater rate of change. The rate of change is determined by the coefficient of \(x\) in the functions.

Please provide the details needed to answer or clarify, and I can help further!