Question

1 m, the total cost, ites. which line represents the total cost, including tax, of an item with a higher tax rate?

1. the graph below represents the amount of money liz borrowed from her father and the number of monthly payments needed to pay it back.

a. what does the slope of the graph represent in the situation?
b. what does the y-intercept in the graph represent in this situation?
c. what is the equation?

1. alyssas carpet cleaning service charges an initial fee of $45, plus $5 for every 100 square feet of carpet cleaned. alyssa graphed y, the amount that her cleaning service charges for x, every 100 square feet of carpet cleaned. what does the y-intercept of alyssas graph represent?

a the charge per square foot of carpet cleaned
b the number of square feet of carpet cleaned
c the total amount owed
d the initial fee

Explanation:

Part 3 (A, B, C) and Part 4

Part 3:

A. Slope Interpretation

Step1: Recall Slope Formula

Slope \( m = \frac{\Delta y}{\Delta x} \). Here, \( y \)-axis: Amount Owed (dollars), \( x \)-axis: Number of Months.

Step2: Analyze Change

From \( (0, 900) \) to \( (10, 150) \) (approx, or use \( (0, 900) \) and \( (10, 0) \) if intercept is 900). Wait, graph shows at \( x=0 \), \( y=900 \); at \( x=10 \), \( y=150 \)? Wait, no, looking at grid: \( x=0 \), \( y=900 \); \( x=10 \), \( y=150 \)? Wait, no, maybe \( x=10 \), \( y=0 \)? Wait, the line goes from (0,900) to (10, 150)? Wait, no, let's check: each grid square is 100? Wait, \( y \)-axis: 0, 100, 200,... 900 at \( x=0 \). At \( x=10 \), \( y=150 \)? No, maybe the end is at \( y=150 \)? Wait, maybe I misread. Alternatively, slope is change in amount owed over change in months. So slope = (Final Owed - Initial Owed)/(Final Months - Initial Months). Initial Owed: 900 (at 0 months), Final Owed: 0 (if it's paid off at 10 months? Wait, the arrow is at \( y \approx 150 \)? Wait, maybe the graph is from (0,900) to (10, 150). Wait, no, let's calculate slope properly. Let's take two points: (0, 900) and (10, 150). Then \( \Delta y = 150 - 900 = -750 \), \( \Delta x = 10 - 0 = 10 \). So slope \( m = \frac{-750}{10} = -75 \). Wait, but maybe the end is at (10, 0)? Then \( \Delta y = 0 - 900 = -900 \), \( \Delta x = 10 - 0 = 10 \), so slope \( m = -90 \). Wait, the graph: at \( x=0 \), \( y=900 \); at \( x=1 \), \( y=800 \) (since from 900 to 800 is 1 grid down). So \( \Delta y = 800 - 900 = -100 \), \( \Delta x = 1 - 0 = 1 \). So slope \( m = -100 \)? Wait, no, \( x=0 \): 900, \( x=1 \): 800, \( x=2 \): 700, so each month, owed decreases by 100? Wait, 900 - 800 = 100 per month. So slope is -100? Wait, maybe the graph is (0,900) to (10, -100)? No, that can't be. Wait, the \( y \)-axis is Amount Owed, so it should decrease as months increase. So slope is (Change in Owed)/(Change in Months) = (Owed at month \( x_2 \) - Owed at month \( x_1 \))/( \( x_2 - x_1 \) ). So if at \( x=0 \), owed is 900; at \( x=1 \), owed is 800, then slope is \( \frac{800 - 900}{1 - 0} = -100 \). So slope represents the monthly payment (since owed decreases by monthly payment each month). So slope is -monthly payment (negative because owed decreases).

Step1: Recall y-intercept

y-intercept is when \( x=0 \) (number of months = 0).

Step2: Analyze \( x=0 \)

At \( x=0 \) (0 months of payments), the amount owed is 900 (from graph: \( y=900 \) when \( x=0 \)). So y-intercept is the initial amount borrowed (since at 0 months, she owes the full borrowed amount).

Step1: Identify Slope and Intercept

From part A, slope \( m = -80 \)? Wait, no, earlier analysis: if at \( x=0 \), \( y=900 \) (y-intercept \( b = 900 \)). From \( x=0 \) to \( x=10 \), \( y \) goes from 900 to 150? No, wait, let's use two points: (0, 900) and (10, 150). Wait, no, looking at the grid: each x (month) increases by 1, y (owed) decreases by 80? Wait, no, from (0,900) to (1,800): decrease by 100. So slope \( m = -100 \). Wait, maybe the graph is (0,900) to (10, -100), but that doesn't make sense. Wait, maybe the correct two points are (0, 900) and (10, 0). Then slope \( m = \frac{0 - 900}{10 - 0} = -90 \). Wait, but the graph shows at \( x=10 \), \( y \approx 150 \). Maybe my initial reading is wrong. Alternatively, let's use the general linear equation \( y = mx + b \). We know \( b = 900 \) (y-intercept). Let's find slope with (0, 900) and (5, 550) (midpoint). \( m = \frac{550 - 900}{5 - 0} = \frac{-350}{5} = -70 \). Hmm, inconsistent. Wait, maybe the graph is (0, 900) to (10, 150), so \( m = \frac{150 - 900}{10 - 0} = -75 \). Then equation is \( y = -75x + 900 \). But let's check with \( x=0 \): \( y=900 \) (correct). \( x=10 \): \( y= -750 + 900 = 150 \) (matches graph). So equation is \( y = -75x + 900 \). Wait, but maybe the slope is -80? Alternatively, maybe the graph is (0, 900) to (10, 0), so \( m = -90 \), equation \( y = -90x + 900 \). But the graph's end is at \( y \approx 150 \), so maybe my grid reading is off. Let's proceed with \( b = 900 \) and slope \( m = -80 \) (approx). Wait, no, let's use exact points. From the graph: \( x=0 \), \( y=900 \); \( x=1 \), \( y=800 \) (since 900 - 100 = 800). So slope \( m = \frac{800 - 900}{1 - 0} = -100 \). Then equation: \( y = -100x + 900 \). Check \( x=1 \): \( y=800 \) (correct). \( x=2 \): \( y=700 \) (correct). Yes, so slope is -100, y-intercept 900.

Step1: Identify \( m \) and \( b \)

\( b = 900 \) (y-intercept, \( x=0 \)). Slope \( m = \frac{800 - 900}{1 - 0} = -100 \).

Step2: Write Equation

Linear equation: \( y = mx + b \), so \( y = -100x + 900 \).

Answer:

The slope represents the monthly payment (amount paid back each month, negative because owed decreases).

B. y-intercept Interpretation