Function Rule F(x) = Mx + B For Plant Food Remaining

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In mathematics, understanding relationships between variables is crucial for solving real-world problems. When dealing with quantities that change over time, we often use functions to model these relationships. This article delves into the process of writing a function rule that describes the relationship between the amount of plant food remaining, denoted as f(x), and the number of days that have passed, represented by x. Our goal is to express this relationship in the form of a linear equation, f(x) = mx + b, where m represents the rate of change (slope) and b represents the initial amount (y-intercept).

Linear functions are fundamental in mathematics and are widely used to model situations where a quantity changes at a constant rate. The general form of a linear function is f(x) = mx + b, where:

  • f(x) is the dependent variable (the amount of plant food remaining in this case).
  • x is the independent variable (the number of days that have passed).
  • m is the slope, representing the rate of change of f(x) with respect to x.
  • b is the y-intercept, representing the value of f(x) when x is zero (the initial amount of plant food).

In the context of plant food remaining over time, the slope m would typically be negative, indicating that the amount of plant food decreases as time passes. The y-intercept b would represent the initial amount of plant food at the beginning.

To determine the function rule f(x) = mx + b for the relationship between plant food remaining and the number of days passed, we need to follow these steps:

  1. Identify Two Points: We need two data points that relate the number of days (x) to the amount of plant food remaining (f(x)). These points can be obtained from a table, a graph, or a word problem. For example, we might know that on day 0, there were 100 grams of plant food, and on day 10, there were 60 grams remaining. This gives us the points (0, 100) and (10, 60).

  2. Calculate the Slope (m): The slope represents the rate at which the plant food is being used up. We can calculate the slope using the formula:

    m = (f(x₂) - f(x₁)) / (x₂ - x₁)

    Using the example points (0, 100) and (10, 60), we have:

    m = (60 - 100) / (10 - 0) = -40 / 10 = -4

    This means that the plant food is decreasing at a rate of 4 grams per day.

  3. Determine the y-intercept (b): The y-intercept is the amount of plant food remaining when x is 0 (the initial amount). If we have a point where x is 0, then the f(x) value of that point is the y-intercept. In our example, the point (0, 100) tells us that the y-intercept b is 100.

    Alternatively, if we don't have a point where x is 0, we can use the slope-intercept form of the equation (f(x) = mx + b) and one of the points to solve for b. For instance, using the point (10, 60) and the slope m = -4, we have:

    60 = -4(10) + b

    60 = -40 + b

    b = 100

  4. Write the Function Rule: Now that we have the slope m and the y-intercept b, we can write the function rule in the form f(x) = mx + b. In our example, with m = -4 and b = 100, the function rule is:

    f(x) = -4x + 100

Let's work through a few example problems to illustrate the process of determining the function rule.

Example 1:

A plant food container initially holds 200 grams. After 15 days, there are 110 grams remaining. Write a function rule to represent the amount of plant food remaining as a function of the number of days passed.

Solution:

  1. Identify Two Points: We have the points (0, 200) and (15, 110).

  2. Calculate the Slope (m):

    m = (110 - 200) / (15 - 0) = -90 / 15 = -6

  3. Determine the y-intercept (b): The point (0, 200) tells us that b = 200.

  4. Write the Function Rule:

    f(x) = -6x + 200

Example 2:

On day 5, there are 75 grams of plant food remaining. On day 20, there are 30 grams remaining. Write a function rule to represent the amount of plant food remaining as a function of the number of days passed.

Solution:

  1. Identify Two Points: We have the points (5, 75) and (20, 30).

  2. Calculate the Slope (m):

    m = (30 - 75) / (20 - 5) = -45 / 15 = -3

  3. Determine the y-intercept (b): We can use the point-slope form of a line (f(x) - f(x₁) = m(x - x₁)) or substitute one of the points into the slope-intercept form (f(x) = mx + b) to solve for b. Let's use the point (5, 75):

    75 = -3(5) + b

    75 = -15 + b

    b = 90

  4. Write the Function Rule:

    f(x) = -3x + 90

Once we have the function rule, it's important to understand what the slope and y-intercept tell us about the situation. In the context of plant food remaining:

  • The slope (m) represents the rate at which the plant food is being used up per day. A negative slope indicates that the amount of plant food is decreasing.
  • The y-intercept (b) represents the initial amount of plant food when no days have passed.

For example, in the function rule f(x) = -4x + 100, the slope of -4 tells us that 4 grams of plant food are being used up each day, and the y-intercept of 100 tells us that there were initially 100 grams of plant food.

Understanding how to write function rules for linear relationships has many practical applications. It allows us to:

  • Predict Future Values: We can use the function rule to predict the amount of plant food remaining after a certain number of days.
  • Determine Initial Conditions: We can find the initial amount of plant food by looking at the y-intercept.
  • Compare Different Scenarios: We can compare the rates at which plant food is being used up in different scenarios by comparing the slopes of their respective function rules.

This concept can be extended to other situations where quantities change linearly over time, such as the depreciation of an asset, the filling of a tank, or the distance traveled at a constant speed.

Writing a function rule for the relationship between the amount of plant food remaining and the number of days passed is a valuable skill in mathematics. By following the steps outlined in this article, you can determine the slope and y-intercept, and then express the relationship in the form of a linear equation. This allows you to model real-world situations, make predictions, and gain a deeper understanding of the changing quantities involved. This understanding extends beyond plant food scenarios, offering a foundational approach to analyzing various linear relationships across different contexts.