Problem 15
Question
(Refer to Example \(1 .\) ) Find either a linear or an exponential function that models the data in the table. $$ \begin{array}{cccccc} x & 0 & 1 & 2 & 3 & 4 \\ \hline y & 2 & 0.8 & -0.4 & -1.6 & -2.8 \end{array} $$
Step-by-Step Solution
Verified Answer
The data is modeled by the linear function \( y = -1.2x + 2 \).
1Step 1: Identify the Type of Function Needed
We need to determine if the data can be modeled by a linear or an exponential function. Let's start by checking if a linear relationship is present. A linear function has a constant rate of change, so we'll calculate the differences between consecutive y-values.
2Step 2: Calculate Differences Between y-values
Calculate the first differences: \[ 0.8 - 2 = -1.2 \]\[ -0.4 - 0.8 = -1.2 \]\[ -1.6 - (-0.4) = -1.2 \]\[ -2.8 - (-1.6) = -1.2 \]Since the first differences are constant (-1.2), the data can be modeled by a linear function.
3Step 3: Find the Linear Function
A linear function can be written as \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. From the differences, the slope \( m = -1.2 \). To find the y-intercept \( b \), plug \( x = 0, y = 2 \) into the equation.\[ 2 = -1.2(0) + b \]\[ b = 2 \]So the linear function is \( y = -1.2x + 2 \).
4Step 4: Verify the Function
Substitute the given values of \( x \) into the linear equation to verify:For \( x = 0 \), \( y = -1.2(0) + 2 = 2 \).For \( x = 1 \), \( y = -1.2(1) + 2 = 0.8 \).For \( x = 2 \), \( y = -1.2(2) + 2 = -0.4 \).For \( x = 3 \), \( y = -1.2(3) + 2 = -1.6 \).For \( x = 4 \), \( y = -1.2(4) + 2 = -2.8 \).All values match the given data, confirming our function.
Key Concepts
Linear RelationshipSlope and InterceptModeling DataRate of Change
Linear Relationship
In mathematics, a linear relationship is one in which two variables move in such a way that they create a straight line when graphed. For a set of data, if the difference between consecutive y-values remains constant, it signals a linear relationship. This means as one variable increases or decreases at a constant rate, the other variable follows a predictable pattern as well.
For example, in our original data set, we calculated the differences between the y-values (such as 0.8 - 2 = -1.2), and these differences remained constant. This confirms a steady change as per our data, which is a firm indicator of a linear relationship. With this confirmation, we can use a linear function to represent the data accurately.
For example, in our original data set, we calculated the differences between the y-values (such as 0.8 - 2 = -1.2), and these differences remained constant. This confirms a steady change as per our data, which is a firm indicator of a linear relationship. With this confirmation, we can use a linear function to represent the data accurately.
Slope and Intercept
The slope and intercept are key components of a linear function, represented by the equation \( y = mx + b \). Here, \( m \) is the slope and \( b \) is the y-intercept. Understanding these terms helps describe the behavior of a linear relationship.
**Slope**
- The slope \( m \) measures the rate of change along the line. It tells us how much \( y \) changes for a one-unit increase in \( x \). For our function, we determined the slope as \( -1.2 \), indicated by the constant differences in y-values.
**Y-intercept**
- The y-intercept \( b \) is the value of \( y \) when \( x = 0 \). It's where the line crosses the y-axis. In our example, when \( x = 0 \), \( y \) was 2, giving us an intercept of 2.
Together, these tell us much about the line's direction and starting point, essential for plotting and understanding linear functions.
**Slope**
- The slope \( m \) measures the rate of change along the line. It tells us how much \( y \) changes for a one-unit increase in \( x \). For our function, we determined the slope as \( -1.2 \), indicated by the constant differences in y-values.
**Y-intercept**
- The y-intercept \( b \) is the value of \( y \) when \( x = 0 \). It's where the line crosses the y-axis. In our example, when \( x = 0 \), \( y \) was 2, giving us an intercept of 2.
Together, these tell us much about the line's direction and starting point, essential for plotting and understanding linear functions.
Modeling Data
Modeling data with functions allows us to make predictions and understand patterns within the data. When data shows a linear relationship, using a linear function makes it easier to work with the data.
In our exercise, once we identified the data's linear nature, we constructed the function \( y = -1.2x + 2 \). Using this function, we can predict y-values for any x-value beyond the ones given in the table.
Modeling data isn't only about plotting points; it's about making informed guesses about future values or understanding current patterns better. This process is crucial in fields such as economics, physics, and biology where data-driven predictions guide decisions.
In our exercise, once we identified the data's linear nature, we constructed the function \( y = -1.2x + 2 \). Using this function, we can predict y-values for any x-value beyond the ones given in the table.
Modeling data isn't only about plotting points; it's about making informed guesses about future values or understanding current patterns better. This process is crucial in fields such as economics, physics, and biology where data-driven predictions guide decisions.
Rate of Change
The rate of change tells us how two variables in a linear function relate to each other. It measures how much one variable changes when the other variable changes by a specific amount.
The rate of change is represented by the slope in a linear equation. For our linear model, the rate of change is \( -1.2 \). This indicates that for each unit increase in \( x \), \( y \) decreases by 1.2 units.
Recognizing the rate of change helps in visualizing the steepness and direction of the line on a graph. A positive rate indicates an upward slope, whereas a negative rate, like in our example, shows a downward slope. This concept is vital in understanding the alignment and scaling of graphs in mathematics.
The rate of change is represented by the slope in a linear equation. For our linear model, the rate of change is \( -1.2 \). This indicates that for each unit increase in \( x \), \( y \) decreases by 1.2 units.
Recognizing the rate of change helps in visualizing the steepness and direction of the line on a graph. A positive rate indicates an upward slope, whereas a negative rate, like in our example, shows a downward slope. This concept is vital in understanding the alignment and scaling of graphs in mathematics.
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