Math Problem Statement

The graph shows the savings in Andre's bank account over time. Calculate the slope and explain its meaning in this context. Determine the vertical intercept and explain what it represents.

Solution

Let's analyze the problem step-by-step.

  1. Slope Calculation (Part a):

    • The slope of a line is calculated by the formula: slope=change in ychange in x\text{slope} = \frac{\text{change in y}}{\text{change in x}}
    • From the graph, we can choose two points. For example, (0, 40) and (10, 80).
    • The change in yy (savings in dollars) is 8040=4080 - 40 = 40.
    • The change in xx (time in weeks) is 100=1010 - 0 = 10.
    • So, the slope is: slope=4010=4\text{slope} = \frac{40}{10} = 4
    • This means that the slope is 4.

    Explanation of the Slope: In this context, the slope represents the rate at which Andre’s savings increase per week. Since the slope is 4, it indicates that Andre’s savings increase by $4 each week.

  2. Vertical Intercept Calculation (Part b):

    • The vertical intercept is the point where the line crosses the y-axis (when x=0x = 0).
    • From the graph, we can see that when x=0x = 0, y=40y = 40.
    • Therefore, the vertical intercept is 40.

    Explanation of the Vertical Intercept: In this context, the vertical intercept represents the initial amount of savings in Andre’s bank account. So, Andre started with $40 in his bank account at week 0.

Would you like more detailed explanations or have any specific questions?


Here are five related questions to further understand the concepts:

  1. How would the graph look if Andre saved $6 per week instead of $4?
  2. What does it mean if the slope of the line is negative in a savings context?
  3. How can we interpret the graph if the line is horizontal?
  4. If Andre started with $50 instead of $40, how would that affect the vertical intercept?
  5. What would be the total savings after 15 weeks if the slope remains the same?

Tip: Remember, the slope in a context like savings can be thought of as the "rate of change" or how fast something is increasing or decreasing.

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Math Problem Analysis

Mathematical Concepts

Linear Equations
Slope and Intercept
Rate of Change

Formulas

slope = (change in y) / (change in x)

Theorems

Slope-Intercept Form of a Line (y = mx + b)

Suitable Grade Level

Grades 6-8