Question

Find a simple function that fits the data in the tables. $$\begin{array}{|r|r|}\hline x & y \\\\\hline-1 & 0 \\\\\hline 0 & 1 \\\\\hline 1 & 2 \\\\\hline 2 & 3 \\\\\hline 3 & 4 \\\\\hline\end{array}$$

Short answer

Answer: The simple function that fits the given data points is y = x + 1.

Step-by-step solution

Calculate the slope (m)

To calculate the slope, we can choose any two points in the given data. From the first two points, (-1, 0) and (0, 1), we can determine the slope as follows: $$m = \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{1 - 0}{0 - (-1)} = \dfrac{1}{1} = 1$$

Calculate the y-intercept (b)

Now that we have the slope, we can use any point in the given data to calculate the y-intercept. Let's use the point (0, 1): $$y = mx + b$$ $$1 = 1 \cdot 0 + b$$ Solving for b, we get: $$b = 1$$

Write the function

Based on our analysis, we now have the slope (m = 1) and the y-intercept (b = 1). The assumed linear function takes the general form y = mx + b. Plugging in the values for m and b, we get the simple function that fits the data: $$y = 1x + 1$$ which can be simplified further as: $$y = x + 1$$

Key concepts

Understanding Slope Calculation

The slope of a line represents the rate of change between the two variables in a linear function. In simpler terms, it tells us how much the variable 'y' changes for each unit increase in 'x'.
To find the slope, we use the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] This formula determines the rise over the run or the vertical change divided by the horizontal change between two points.
For example, using the points (-1, 0) and (0, 1) from the data table, plug these values into the slope formula:

• Change in y : 1 - 0 = 1
• Change in x : 0 - (-1) = 1

This gives us a slope (m) of 1, meaning that for every increase of 1 in 'x', 'y' also increases by 1. This linearity reflects a constant relationship between 'x' and 'y'.
The slope is an essential part of defining the equation of a line as it determines its steepness and direction.

Finding the Y-Intercept

The y-intercept is where the line crosses the y-axis on a graph. It gives the value of 'y' when 'x' equals zero. In the equation of a line, often expressed as \( y = mx + b \), the 'b' represents the y-intercept.
Using the point (0, 1) from the data table, it's straightforward to see that when 'x' is 0, 'y' is 1. This means our y-intercept (b) is 1.
Here's how it is derived from the equation using the calculated slope:

• With the slope \( m = 1 \) and substituting \( x = 0 \), the equation looks like: \( y = 1 \cdot 0 + b \)
• Solving: \( 1 = 0 + b \) leads to \( b = 1 \)

Calculating the y-intercept confirms where the line crosses the y-axis and is crucial in forming the complete equation of the line.

Applying Data Fitting in Linear Functions

Data fitting in the context of linear functions involves finding an equation that best represents the dataset provided. By determining both the slope and y-intercept, we can construct a simple linear model to explain the relationship between variables.
From the given data,

• Calculated slope \( m = 1 \)
• Calculated y-intercept \( b = 1 \)

The equation formed is \( y = x + 1 \), reflecting a fitted linear model.
This model suggests a direct, proportional relationship: for each unit increase in 'x', 'y' increases by the same unit.
Creating such a model is useful in various applications, like predicting trends or analyzing the relationship between variables. It's an essential step for understanding and extracting meaningful insights from data sets.