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Magazine Chapter 1: Problem 30
Determine whether each of the following tables of values could correspond to a linear function, an exponential function, or neither. For each table of values that could correspond to a linear or an exponential function, find a formula for the function. A. $$\begin{array}{c|c} \hline x & f(x) \\ \hline 0 & 10.5 \\ 1 & 12.7 \\ 2 & 18.9 \\ 3 & 36.7 \\ \hline \end{array}$$ B. $$\begin{array}{c|l} \hline t & s(t) \\ \hline-1 & 50.2 \\ 0 & 30.12 \\\ 1 & 18.072 \\ 2 & 10.8432 \\ \hline \end{array}$$ C. $$\begin{array}{c|c} \hline u & g(u) \\ \hline 0 & 27 \\ 2 & 24 \\ 4 & 21 \\ 6 & 18 \\ \hline \end{array}$$
Short Answer
Expert verified
Table A is neither. Table B is exponential: \( s(t) = 30.12 (0.6)^t \). Table C is linear: \( g(u) = -1.5u + 27 \).
Step by step solution
01
Analyze Table A for Linearity
To determine if Table A corresponds to a linear function, check for a consistent difference in the function values \( f(x) \) as \( x \) increases by 1.- \( f(1) - f(0) = 12.7 - 10.5 = 2.2 \)- \( f(2) - f(1) = 18.9 - 12.7 = 6.2 \)- \( f(3) - f(2) = 36.7 - 18.9 = 17.8 \)The differences are not consistent, so it is not a linear function.
02
Check Table A for Exponential Function
To evaluate if Table A corresponds to an exponential function, check if there is a consistent ratio between consecutive terms.- \( \frac{f(1)}{f(0)} = \frac{12.7}{10.5} \approx 1.21 \)- \( \frac{f(2)}{f(1)} = \frac{18.9}{12.7} \approx 1.49 \)- \( \frac{f(3)}{f(2)} = \frac{36.7}{18.9} \approx 1.94 \)The ratios are inconsistent, so Table A does not represent an exponential function.
03
Analyze Table B for a Linear Function
Examine Table B for a possible linear relationship by checking for consistent differences in \( s(t) \) values as \( t \) increases by 1.- \( s(0) - s(-1) = 30.12 - 50.2 = -20.08 \)- \( s(1) - s(0) = 18.072 - 30.12 = -12.048 \)- \( s(2) - s(1) = 10.8432 - 18.072 = -7.2288 \)The differences are not consistent, so it is not linear.
04
Check Table B for Exponential Function
Analyze Table B for an exponential relationship by examining the ratio of consecutive terms.- \( \frac{s(0)}{s(-1)} = \frac{30.12}{50.2} \approx 0.6 \)- \( \frac{s(1)}{s(0)} = \frac{18.072}{30.12} \approx 0.6 \)- \( \frac{s(2)}{s(1)} = \frac{10.8432}{18.072} \approx 0.6 \)The ratios are consistent, suggesting an exponential function with decay factor 0.6.
05
Determine Formula for Table B
For an exponential function, the formula follows \( s(t) = a \, b^t \), where \( a = s(0) = 30.12 \) and \( b = 0.6 \).Thus, the function is \( s(t) = 30.12 \, (0.6)^t \).
06
Analyze Table C for a Linear Function
Check Table C if it fits a linear function by evaluating consistent differences in \( g(u) \) over equal intervals of \( u \).Differences over intervals of 2 units:- \( g(2) - g(0) = 24 - 27 = -3 \)- \( g(4) - g(2) = 21 - 24 = -3 \)- \( g(6) - g(4) = 18 - 21 = -3 \)The differences are the same, confirming it is a linear function.
07
Determine Formula for Table C
The linear function formula is \( g(u) = mu + c \) where the slope \( m \) is found using \( m = \frac{g(2) - g(0)}{2} = \frac{-3}{2} = -1.5 \).Using point (0, 27), solve for \( c \):\(27 = -1.5(0) + c\), so \(c = 27\).Thus, \( g(u) = -1.5u + 27 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Linear Function
A linear function represents a straight-line relationship between two variables. When you have a table of values and want to check if they form a linear function, you look for consistent changes in the values. This means that for each unit increase in the input (say, \(x\)), the output (say, \(f(x)\)) should change by the same amount. This is called a constant difference or rate of change.
For instance, if you're analyzing a table and find that each time the input goes up by one, the output goes up (or down) by the same amount, then you have a linear function. If you're given that \(g(u) = -1.5u + 27\), this formula gives you a linear function because it shows that for each increase in \(u\), \(g(u)\) decreases consistently by \(-1.5\). The graph of this function would be a straight line with a negative slope.
For instance, if you're analyzing a table and find that each time the input goes up by one, the output goes up (or down) by the same amount, then you have a linear function. If you're given that \(g(u) = -1.5u + 27\), this formula gives you a linear function because it shows that for each increase in \(u\), \(g(u)\) decreases consistently by \(-1.5\). The graph of this function would be a straight line with a negative slope.
- Constant change in outputs suggests linearity.
- The function鈥檚 graph is a straight line: slope is constant.
- Formula form: \(y = mx + c\) where \(m\) is slope, \(c\) is y-intercept.
Exponential Function
Exponential functions differ from linear functions in that rate of change happens by a constant ratio, rather than a difference. To identify these from a table of values, look for a consistent ratio between consecutive outputs for a constant increase in the inputs. This type of function is often used to model growth or decay processes.
If, for example, you have a table and observe that with each increase of 1 in the input \( t \), the output \( s(t) \) is multiplied by a consistent factor, you are dealing with an exponential function. In this problem, \( s(t) = 30.12 (0.6)^t \) illustrates such a relationship, where each output \(s(t)\) results from the prior one multiplied by 0.6, showing exponential decay.
If, for example, you have a table and observe that with each increase of 1 in the input \( t \), the output \( s(t) \) is multiplied by a consistent factor, you are dealing with an exponential function. In this problem, \( s(t) = 30.12 (0.6)^t \) illustrates such a relationship, where each output \(s(t)\) results from the prior one multiplied by 0.6, showing exponential decay.
- Constant ratio between outputs is a key sign.
- Often models growth (above 1) or decay (below 1).
- Formula form: \(y = ab^x\), where \(b\) is the base of the exponential.
Function Analysis
Function analysis involves identifying patterns in tables of values to determine the type of function they represent. This process includes looking for consistent differences indicative of linear functions or consistent ratios indicative of exponential functions. This requires careful examination of changes between outputs as inputs change.
In the given exercise, analyzing patterns helps solve parts of the exercise. For example, in table C, a consistent difference of \(-3\) over 2-unit gaps suggests a linear function. Whereas, in table B, a pattern of reducing outputs by a factor of 0.6 suggests an exponential function.
In the given exercise, analyzing patterns helps solve parts of the exercise. For example, in table C, a consistent difference of \(-3\) over 2-unit gaps suggests a linear function. Whereas, in table B, a pattern of reducing outputs by a factor of 0.6 suggests an exponential function.
- Identify linear functions: look for constant differences.
- Identify exponential functions: look for constant ratios.
- Analyze interval changes for consistent results.
Mathematical Tables
Mathematical tables present data pairs that reveal the relationship between two variables. Tables are crucial for detecting function types when the relationships are not immediately obvious. They offer a structured way to observe patterns, whether differences or ratios, and base decisions about function types on these observations.
Consider the tables in your exercise; each provides different data to test for potential function types. By carefully computing differences or ratios between values, you can often discern whether the table suggests a linear or exponential function, or possibly neither.
Consider the tables in your exercise; each provides different data to test for potential function types. By carefully computing differences or ratios between values, you can often discern whether the table suggests a linear or exponential function, or possibly neither.
- Use tables to see changes clearly between data points.
- Calculate differences or ratios for pattern recognition.
- Base function type conclusions on systematic observations.
Function Formula
The function formula expresses the relationship between input and output numerically. It allows you to predict future outputs or backtrack to past inputs, given the nature of the function. Derived typically from patterns in data, once a function type is identified, you formulate the mathematical expression.
For linear functions, the formula uses the slope-intercept form \( y = mx + c \). For example, \( g(u) = -1.5u + 27 \) directly follows from identified patterns of change. Meanwhile, exponential functions turn patterns into expressions like \( y = ab^x \), as seen in \( s(t) = 30.12 \, (0.6)^t \), derived from a consistent decay rate.
For linear functions, the formula uses the slope-intercept form \( y = mx + c \). For example, \( g(u) = -1.5u + 27 \) directly follows from identified patterns of change. Meanwhile, exponential functions turn patterns into expressions like \( y = ab^x \), as seen in \( s(t) = 30.12 \, (0.6)^t \), derived from a consistent decay rate.
- Once identified, create formulas using pattern insights.
- Linear functions: straight-line formulas \( y = mx + c \).
- Exponential functions: growth/decay formulas \( y = ab^x \).
