x y

Question

-2 80
-1 70
2 40
5 10
Answer the questions using the table.

(3 points)
the rate of change =


the y−intercept =

the equation of the line in slope intercept form is


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Answer 1

To analyze the data in the table, we can determine the rate of change (slope), the y-intercept, and form an equation in slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept.

1. Rate of change (slope): The rate of change (slope) can be calculated using two points from the table. Let's use the points (-1, 70) and (2, 40):

   \[ \text{slope} = m = \frac{y_2 - y_1}{x_2 - x_1} \]

   Using the points (x₁, y₁) = (-1, 70) and (x₂, y₂) = (2, 40):

   \[ m = \frac{40 - 70}{2 - (-1)} = \frac{-30}{2 + 1} = \frac{-30}{3} = -10 \]

   So, the rate of change (slope) is -10.

2. Y-intercept: To find the y-intercept, we can use the point-slope form of the equation. We know the slope is -10 and can use one of the points, say (2,40):

   Using the slope-intercept formula \(y = mx + b\):

   \[ 40 = -10(2) + b \] \[ 40 = -20 + b \] \[ b = 40 + 20 = 60 \]

   Therefore, the y-intercept is 60.

3. Equation of the line in slope-intercept form: Now that we have the slope (m = -10) and the y-intercept (b = 60), we can write the equation:

   \[ y = -10x + 60 \]

In summary:

• The rate of change = -10
• The y-intercept = 60
• The equation of the line in slope-intercept form is y = -10x + 60.

Answer 2

Domain considering the scenario: Justin has $10 in his account and plans to save $25 each week until he reaches $135 for the new game he wants to buy.

Answer 3

To determine the domain of Justin's savings scenario, we first need to understand the context and establish the variables involved.

1. Initial Conditions:

   • Justin starts with $10 in his account.
   • He plans to save $25 each week.
   • He wants to reach a total of $135.

2. Setting Up the Equation: Let \( x \) represent the number of weeks he saves.

The total amount \( A \) in his account after \( x \) weeks can be expressed as: \[ A = 10 + 25x \]

Justin wants this total amount \( A \) to be at least $135 to buy the game: \[ 10 + 25x \geq 135 \]

3. Solving the Inequality: Subtracting 10 from both sides gives: \[ 25x \geq 125 \]

Dividing by 25: \[ x \geq 5 \]

This means that Justin needs to save for at least 5 weeks to reach his goal.

4. Determining the Domain: Since \( x \) represents weeks, and he can't save for a negative number of weeks, the natural boundaries for \( x \) would be:

• Lower Bound: \( x \geq 0\) (he can start from week 0)
• Upper Bound: There is no upper limit since he can continue to save even after reaching $135.

Considering the context:

• The lowest he can start is week 0.
• He needs at least 5 weeks to save enough money.

Thus, the valid domain for \( x \) in this scenario is: \[ [5, \infty) \]

In conclusion, the domain of the number of weeks \( x \) that Justin can save until he reaches his goal is [5, \infty).