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?

Question

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.

Answer 1

Let's go through your questions one by one.

Question 1

To solve the equation \(10 - x^2 = x + 4\), we can rearrange it to equate both sides to zero: \[ 10 - x^2 - x - 4 = 0 \ -x^2 - x + 6 = 0 \ x^2 + x - 6 = 0 \] This can be solved by graphing the following two functions:

• \(f(x) = 10 - x^2\) (the left side)
• \(g(x) = x + 4\) (the right side)

Finding the intersection points of these two graphs will give the solution to the original equation. The solution set is where these two functions equate:

• The correct response is: g(x) = x + 4, f(x) = 10 - x^2, {−3, 2} (indicating the points where the two functions intersect).

Question 2

To solve the equation \(9x - 6 = -8\), we can rearrange it to isolate the variable: \[ 9x - 6 + 6 = -8 + 6 \ 9x = -2 \ x = -\frac{2}{9} \] To graphically find this solution, we should express one side of the equation in terms of \(f(x)\):

• The correct function is: f(x) = 9x - 14 (which expresses the left side of the equation equal to -8).

Question 3

For the equation \(2x - 1 = 4x + 3\), we can solve for \(x\): \[ 2x - 1 - 4x = 3 \ -2x - 1 = 3 \ -2x = 3 + 1 \ -2x = 4 \ x = -2 \] Thus, the solution is: \[ x = -2 \]

Question 4

In this problem:

• Your rate of change (how fast you are walking toward home) is 3 blocks per minute.
• Your starting amount (total distance from your house) is 41 blocks.
• The equation that represents the situation in slope-intercept form can be expressed as: \[ y = -3x + 41 \] where \(y\) is the distance to home and \(x\) is the time in minutes.

To find how long it will take to get home: Setting \(y = 0\) (when you reach home), \[ 0 = -3x + 41 \ 3x = 41 \ x = \frac{41}{3} \approx 13.67 \text{ minutes} \approx 14 \text{ minutes (to the nearest minute)} \]

Summary of Answers

1. g(x) = x + 4, f(x) = 10−x^2, {−3, 2}
2. f(x) = 9x - 14
3. x = -2
4. • Your rate of change: 3
   • Your starting amount: 41
   • Equation in slope-intercept form: \(y = -3x + 41\)
   • It will take: 14 minutes