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Chapter 1: Problem 1

(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 is \( f(x) = 2x - 3 \).

Step by step solution

01

Finding the Family of Linear Functions with Slope 2

A linear function can be represented as \( f(x) = mx + c \), where \( m \) is the slope. Given the slope of 2, the general equation becomes \( f(x) = 2x + c \). This equation represents a family of linear functions with varying \( c \) values, determining the different y-intercepts.
02

Sketching Members of the Family

To sketch members of the family, select different values for \( c \) such as 0, 1, and -1. For \( c = 0 \), the function is \( f(x) = 2x \). For \( c = 1 \), it's \( f(x) = 2x + 1 \), and for \( c = -1 \), it's \( f(x) = 2x - 1 \). The graphs will be parallel lines with the same slope but different y-intercepts.
03

Finding the Family of Linear Functions where f(2) = 1

This condition means \( f(2) = 2m + c = 1 \). We rearrange to find \( c = 1 - 2m \). Thus, the equation for this family is \( f(x) = mx + (1 - 2m) \). This represents a set of lines crossing the point (2, 1), differing by slope \( m \).
04

Sketching Members of this Family

Choose different slopes, say \( m = 0 \), \( m = 1 \), and \( m = -1 \). For \( m = 0 \), \( f(x) = 1 \). For \( m = 1 \), \( f(x) = x - 1 \). For \( m = -1 \), \( f(x) = -x + 3 \). These lines will all cross the point (2, 1).
05

Finding the Intersection of Both Families

A function belonging to both families must satisfy both conditions: a slope of 2 and passing through (2, 1). From the second family equation, substitute \( m = 2 \) into \( c = 1 - 2m \), yielding \( c = 1 - 4 = -3 \). Thus, the function is \( f(x) = 2x - 3 \).
06

Verify and Sketch the Intersection Function

Verification: This function has a slope of 2, and substituting \( x = 2 \), we find \( f(2) = 2(2) - 3 = 1 \), confirming the point (2, 1). Graphing \( f(x) = 2x - 3 \) shows it satisfies both criteria.

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Key Concepts

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

Understanding the Slope of a Line
The slope of a line is a measure of its steepness. It tells us how much the line rises or falls as it moves from left to right. In the formula for a linear function, written as \( f(x) = mx + c \), the variable \( m \) represents the slope.
  • If the slope \( m \) is positive, the line rises as you move from left to right.
  • If \( m \) is negative, the line falls as you move from left to right.
  • A zero slope means the line is horizontal.
  • An undefined slope (dividing by zero in the calculation of slope) means the line is vertical.
Understanding the concept of slope helps us predict the direction and angle of the line based on its value.
Grasping the Y-Intercept
The y-intercept is the point where the line crosses the y-axis. In the equation \( f(x) = mx + c \), the y-intercept is represented by \( c \). This is the value of \( y \) when \( x \) is zero. Knowing the y-intercept helps us understand where the line sits vertically on a graph.When drawing a linear function, the y-intercept gives you a starting point. You can then use the slope to determine the direction and steepness of the line from that point. The y-intercept is crucial for sketching the graph of a linear function, which can be different for every member of a family of lines.
Exploring Point-Slope Form
The point-slope form is another way to write the equation of a line when you know the slope and a specific point on the line. The formula is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line.This form is particularly useful when you know certain specific points that the line must pass through, such as \( f(2) = 1 \) in the problem presented. It allows you to quickly find the equation of a line by adjusting the slope and translating through different points.With the point-slope form, you can easily apply real-world data, finding the proper line equations that meet given conditions.
Understanding Parallel Lines
Parallel lines have the same slope but different y-intercepts. This means they never intersect, remaining equidistant forever. In practical terms, two lines \( y = mx + c_1 \) and \( y = mx + c_2 \) are parallel if both have the same slope \( m \), but different intercepts \( c_1 \) and \( c_2 \).This concept was demonstrated in the exercise where you have a family of lines all with a slope of 2 but varying y-intercepts. When graphing, these lines appear as parallel, reinforcing our understanding of how the slope and y-intercept influence the position of a line.Understanding the principle of parallelism in lines is crucial in geometry and various applications, ensuring precise calculations and system interpretations.

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