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Magazine Chapter 5: Problem 5
Construct both a linear and an exponential function that go through the points (0,6) and (1,9) .
Short Answer
Expert verified
Linear function: \( y = 3x + 6 \). Exponential function: \( y = 6(1.5)^x \).
Step by step solution
01
Identify the Form of the Linear Function
The linear function will be in the form of \( y = mx + b \). Here, \( m \) is the slope of the line and \( b \) is the y-intercept.
02
Calculate the Slope
Use the formula for the slope \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using points \((0, 6)\) and \((1, 9)\), \( m = \frac{9 - 6}{1 - 0} = 3 \).
03
Determine the Y-Intercept
Substitute one of the points into the linear equation to find the y-intercept \( b \). Using the point \((0, 6)\), which is the y-intercept itself, we find \( b = 6 \).
04
Form the Linear Equation
Using \( m = 3 \) and \( b = 6 \), the linear function is \( y = 3x + 6 \).
05
Identify the Form of the Exponential Function
The exponential function will be in the form of \( y = ab^x \). Here, \( a \) and \( b \) are constants.
06
Determine the Value of \( a \)
Using the point \((0, 6)\), substitute \( x = 0 \) into the exponential function. Since \( y = 6 \), this gives \( 6 = ab^0 = a \). Therefore, \( a = 6 \).
07
Determine the Value of \( b \)
Using the point \((1, 9)\), substitute \( x = 1 \) into the exponential function. With \( y = 9 \), this gives \( 9 = 6b \). Solving for \( b \) yields \( b = \frac{9}{6} = 1.5 \).
08
Form the Exponential Equation
Using \( a = 6 \) and \( b = 1.5 \), the exponential function is \( y = 6(1.5)^x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slope
A key concept in understanding linear functions is the slope, denoted by the letter \(m\). The slope represents the rate of change between two variables. To put it simply, it tells us how much the \(y\)-value changes for every one unit increase in the \(x\)-value.
In the given exercise, we use the points \((0, 6)\) and \((1, 9)\) to find the slope. The formula for the slope is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Plugging in our points, we get \(m = \frac{9 - 6}{1 - 0} = 3\).
So, the slope \(m\) equals 3, meaning for every one unit increase in \(x\), the \(y\)-value increases by 3 units.
Knowing the slope helps us form the linear equation, integrating the change rate directly into our function.
In the given exercise, we use the points \((0, 6)\) and \((1, 9)\) to find the slope. The formula for the slope is \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Plugging in our points, we get \(m = \frac{9 - 6}{1 - 0} = 3\).
So, the slope \(m\) equals 3, meaning for every one unit increase in \(x\), the \(y\)-value increases by 3 units.
Knowing the slope helps us form the linear equation, integrating the change rate directly into our function.
Y-Intercept
Next, let's focus on the y-intercept, denoted by the letter \(b\). The y-intercept is the point where the line crosses the \(y\)-axis. This occurs when \(x = 0\).
In our exercise, one of the given points is \((0, 6)\). This automatically gives us the y-intercept value \(b = 6\).
To form the linear equation, we combine the slope and the y-intercept. With the slope \(m = 3\) and the y-intercept \(b = 6\), we get the linear equation as \(y = 3x + 6\).
The y-intercept helps us quickly determine the starting point of the function on the \(y\)-axis. This makes it easier to graph and understand the behavior of the linear function from its origin.
In our exercise, one of the given points is \((0, 6)\). This automatically gives us the y-intercept value \(b = 6\).
To form the linear equation, we combine the slope and the y-intercept. With the slope \(m = 3\) and the y-intercept \(b = 6\), we get the linear equation as \(y = 3x + 6\).
The y-intercept helps us quickly determine the starting point of the function on the \(y\)-axis. This makes it easier to graph and understand the behavior of the linear function from its origin.
Exponential Function
Understanding exponential functions is crucial as well. An exponential function has the general form \(y = ab^x\), where \(a\) and \(b\) are constants. The \(a\) value is the initial amount or the y-intercept for \(x = 0\), and \(b\) is the base of the exponential, representing the growth factor.
To find \(a\), we use the point \((0, 6)\). Given \(y = 6\) when \(x = 0\), substituting these values gives us \(6 = ab^0 \). Since \(b^0 = 1\), we have \(a = 6\).
Next, to find \(b\), we use the point \((1, 9)\). Substituting into the exponential form, \(9 = 6b^1\). Solving for \(b\) gives us \(b = \frac{9}{6} = 1.5\).
Therefore, the exponential function is \( y = 6(1.5)^x \).
Exponential functions are powerful in modeling growth patterns, such as population growth, where values increase by a consistent percentage rather than a fixed amount.
To find \(a\), we use the point \((0, 6)\). Given \(y = 6\) when \(x = 0\), substituting these values gives us \(6 = ab^0 \). Since \(b^0 = 1\), we have \(a = 6\).
Next, to find \(b\), we use the point \((1, 9)\). Substituting into the exponential form, \(9 = 6b^1\). Solving for \(b\) gives us \(b = \frac{9}{6} = 1.5\).
Therefore, the exponential function is \( y = 6(1.5)^x \).
Exponential functions are powerful in modeling growth patterns, such as population growth, where values increase by a consistent percentage rather than a fixed amount.
