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Chapter 1: Problem 5

(a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family. (b) Find an equation for the family of linear functions such that \(f(2)=1\) and sketch several members of the family. (c) Which function belongs to both families?

Short Answer

Expert verified
The function \( f(x) = 2x - 3 \) belongs to both families.

Step by step solution

01

Equation of the Family of Linear Functions with Slope 2

A linear function can be expressed in the form \( f(x) = mx + c \), where \( m \) is the slope. Given that the slope \( m = 2 \), the equation for the family of linear functions is \( f(x) = 2x + c \). Here, \( c \) represents any constant, which means there are infinitely many parallel lines with slope 2 that make up this family.
02

Equation of the Family of Linear Functions through Point (2,1)

For the second family, the function must satisfy \( f(2) = 1 \). By substituting into the linear function \( f(x) = mx + c \), we have \[ 1 = m(2) + c \]\[ 1 = 2m + c \]This equation must hold for any function in this family. To find a specific form, assume any value of \( m \) and solve for \( c \). Generally, the family is defined by \( c = 1 - 2m \), giving the family of functions as \( f(x) = mx + (1 - 2m) \).
03

Find the Common Function in Both Families

A function belongs to both families if it has a slope of 2 (from Step 1) and passes through point (2,1) (from Step 2). Using the equation for a family of functions with slope 2, \( f(x) = 2x + c \), substitute \( x = 2 \) and \( f(x) = 1 \):\[ 1 = 2(2) + c \]Solving for \( c \), we get:\[ 1 = 4 + c \]\[ c = -3 \]Thus, the function \( f(x) = 2x - 3 \) belongs to both families.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope
The concept of a slope is fundamental to understanding linear functions. The slope measures how steep a line is on a graph and is often represented by the letter \( m \) in the linear equation \( f(x) = mx + c \). A positive slope means that the line rises as it moves from left to right, while a negative slope means it falls.
  • If the slope is 0, the line is perfectly horizontal, indicating no change in y for a change in x.
  • When the slope is undefined, the line is vertical, reflecting an infinite change in y for a small change in x.
For our exercise, the slope given is 2, meaning the line rises by 2 units for every 1 unit increase in the x-direction. This constant rate of increase is what defines the linear relationship.
Linear Equations
A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. The general form of a linear equation is \( f(x) = mx + c \), where\( m \) is the slope and \( c \) is the y-intercept.
  • The y-intercept, \( c \), is the point where the line crosses the y-axis. It indicates the value of \( f(x) \) when \( x = 0 \).
  • Linear equations are powerful tools for modeling relationships that have a constant rate of change.
In our exercise, understanding the linear equation helps in visualizing the family of linear functions with given characteristics, such as a specific slope or passing through a particular point.
Functions and Graphs
Functions and their graphs are visual representations of mathematical relationships between variables. For linear functions, their graphs are straight lines, making them simple but very informative.
  • A graph of a linear function \( f(x) = mx + c \) is a straight line, where the slope \( m \) determines the tilt of the line, and the y-intercept \( c \) ensures where it cuts through the y-axis.
  • Graphs help visually compare different linear functions by showing how steeply they rise or fall and where they start on the y-axis.
In the exercise, sketching the graphs of various functions from the families helps to easily see the differences and similarities, illustrating concepts such as parallel lines and intersection points.
Intersecting Lines
Intersecting lines are lines that cross each other at some point. In the context of linear functions, two distinct lines will intersect if they have different slopes. The point where they meet is called the intersection point.
  • Lines that are parallel (same slope) will never intersect.
  • If two lines are part of distinct families but intersect, it means they share at least one common point.
In our exercise, the task was to find a function that belonged to both families. This is effectively finding when two lines, one from each family, would intersect. The only way this is possible is if they are the exact same line, which was shown for \( f(x) = 2x - 3 \). Here, the same slope and meeting point (intersection) fulfill the requirement for intersection.

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Most popular questions from this chapter

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 \(\mathrm{cm} / \mathrm{s}\) . (a) Express the radius \(r\) of this circle as a function of the time \(t(\) in seconds). (b) If \(A\) is the area of this circle as a function of the radius,find \(A \circ r\) and interpret it.

Find a formula for the inverse of the function. \(f(x)=e^{2 x-1}\)

(a) Suppose \(f\) is a one-to-one function with domain \(A\) and range \(B .\) How is the inverse function \(f^{-1}\) defined? What is the domain of \(f^{-1} ?\) What is the range of \(f^{-1} ?\) (b) If you are given a formula for \(f,\) how do you find a formula for \(f^{-1} ?\) (c) If you are given the graph of \(f,\) how do you find the graph of \(f^{-1} ?\)

Data points \((x, y)\) are given. (a) Draw a scatter plot of the data points. (b) Make semilog and log-log plots of the data. (c) Is a linear, power, or exponential function appropriate for modeling these data? (d) Find an appropriate model for the data and then graph the model together with a scatter plot of the data. $$\begin{array}{|c|c|c|c|c|c|c|}\hline x & {2} & {4} & {6} & {8} & {10} & {12} \\\ \hline y & {0.08} & {0.12} & {0.18} & {0.26} & {0.35} & {0.53} \\\ \hline\end{array}$$ \(\begin{array}{|c|c|c|c|c|c|c|}\hline x & {0.5} & {1.0} & {1.5} & {2.0} & {2.5} & {3.0} \\ \hline y & {4.10} & {3.71} & {3.39} & {3.2} & {2.78} & {2.53} \\ \hline\end{array}\)

Drinking and driving In a medical study, researchers measured the mean blood alcohol concentration (BAC) of eight fasting adult male subjects (in mg/mL) after rapid consumption of 30 \(\mathrm{mL}\) of ethanol (corresponding to two standard alcoholic drinks). The BAC peaked after half an hour and the table shows measurements starting after an hour. $$\begin{array}{|c|c|c|c|c|c|}\hline t \text { hours) } & {1.0} & {1.25} & {1.5} & {1.75} & {2.0} \\ \hline \mathrm{BAC} & {0.33} & {0.29} & {0.24} & {0.22} & {0.18} \\ \hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|c|}\hline \text { t(hours) } & {2.25} & {2.5} & {3.0} & {3.5} & {4.0} \\ \hline \mathrm{BAC} & {0.15} & {0.12} & {0.069} & {0.034} & {0.010} \\ \hline\end{array}$$ (a) Make a scatter plot and a semilog plot of the data. (b) Find an exponential model and graph your model with the scatter plot. Is it a good fit? (c) Use your model and logarithms to determine when the BAC will be less than 0.08 \(\mathrm{mg} / \mathrm{mL}\) , the legal limit for driving.
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