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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−x^2=x+4

(1 point)
Responses

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

g(x) = x + 4, f(x) = 10+x^2, {−3,1}
g(x) = x + 4, f(x) = 10+x^2, {−3,1}

g(x) = x + 4, f(x) = 10−x^2, {1,6}
g(x) = x + 4, f(x) = 10−x^2, {1,6}

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

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

(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?

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(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:

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(1 point)
Function
has a greater rate of change.
Question 13
12.

Salim purchased a motorcycle and it depreciates each year. Its depreciation is represented by the function f(x) = 14000(1−0.09)^x

When will the motorcycle be worth $7000?

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(1 point)
Responses

6.7 years
6.7 years

7 years
7 years

7.3 years
7.3 years

8 years
8 years

1 answer

Answers to Questions

Question 1: To solve the equation \(10 - x^2 = x + 4\), we can rearrange it to two functions:

  • \(f(x) = 10 - x^2\)
  • \(g(x) = x + 4\)

Using graphing methods, the solution set where they intersect corresponds to:

  • Response: \(g(x) = x + 4, f(x) = 10 - x^2, {-3, 2}\)

Question 2: To solve the equation \(9x - 6 = -8\), we can rearrange it into a function:

  • \(f(x) = 9x + 2\) (since \(-8\) becomes \(9x - 6 + 8 = 0\))

  • Response: \(f(x) = 9x + 2\)

Question 3: To solve \(2x - 1 = 4x + 3\): \[ 2x - 4x = 3 + 1 \implies -2x = 4 \implies x = -2 \]

  • Answer: x = -2

Question 4: The situation describes walking toward home at a constant rate. The variable \(y\) can describe the distance. Therefore:

  1. Your rate of change: 3 (blocks per minute)
  2. Your starting amount: 41 (blocks)
  3. The equation representing the situation in slope-intercept form is: \[y = -3x + 41\]
  4. To find the time taken to get home: \[ 0 = -3x + 41 \rightarrow 3x = 41 \rightarrow x = \frac{41}{3} \approx 14 \text{ minutes (rounded to the nearest minute)} \]

Question 5: For the water tank scenario:

  1. Initial amount (how much water the tank had when it started being drained): (initial volume)
  2. The tank is empty after: (time until empty)
  3. The rate of emptying: (rate)
  4. The equation that would represent this scenario would start with the initial amount and decrease at the given rate.

(Details will depend on the specific slope and data provided.)

Question 6: To find when Ella catches up with her mother, you can graph both distances:

  • Brandi's equation: \(y = 4x\)

  • Ella's equation: \(y = 6x - 12\)

  • Response: Line 1 and Line 2

Question 7: For the decay of the radioactive substance:

  • The correct exponential function modeling decay of 6%: \[f(x) = 600(0.94)^x\]
  • Response: \(f(x) = 600(0.94)^x\)

Question 8: Assuming actual value provided for the species count:

  • Answer: The starting amount was (initial species count).

Question 9: For the golf ball:

  1. This scenario would be represented by a quadratic equation.
  2. The ball lands 60 feet away, provides the x-intercept. Half the distance serves as the axis of symmetry. The vertex is at (30, 30).
  3. It starts at 0 feet away from Lorraine; thus another point could be (0, 0).
  4. Value that needs to be found is the height of the ball in terms of some variable.

Question 10: To find the height of the golf ball when it's 15 feet away:

  1. Set the quadratic function into standard form, find the values.
  2. The equation's form would relate to its vertex and points accordingly.
  3. The height: \( (calculated height) \).

Question 11: To find when the rock hits the bottom:

  1. Set the function \(f(x) = -4.9x^2 + 9\) to -2 and solve for \(x\): \[ -2 = -4.9x^2 + 9 \implies -4.9x^2 = -11 \implies x^2 = \frac{11}{4.9} \implies x = \sqrt{\frac{11}{4.9}} \]
  • Answer: The rock will hit at around \( 1.5 \) seconds (to the nearest tenth of a second).

Question 12: Examining function rates, compare their slopes:

  1. Function 1 \(f(x) = 32x + 5\) has a slope of 32.
  2. Compare it with other functions provided in context.

Question 13: To find when the motorcycle is worth $7000: Set the equation \(14000(1 - 0.09)^x = 7000\). Solve for \(x\):

  1. \( \frac{7000}{14000} = (0.91)^x \), logarithmic solution would determine \(x\).
  • Response: (based on calculated years)

Overall, for each question, ensure calculations are precise and based on the principles of linear and quadratic functions, while also utilizing graphing for intersection points.