Problem 14 - Precalculus Mathematics for Calculus

Question

(a) Sketch lines through \((0,0)\) with slopes \(1,0, \frac{1}{2}, 2\) and \(-1\) (b) Sketch lines through \((0,0)\) with slopes \(\frac{1}{3}, \frac{1}{2},-\frac{1}{3},\) and 3.

Step-by-Step Solution

Answer

Sketch lines from origin for given slopes, using \( y = mx \) for each. Slope determines line steepness and orientation.

Step 1: Understand Slope-Intercept Form

The slope-intercept form of a line is given by \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. Lines passing through the origin \((0,0)\) will have \( c = 0 \), so their equations become \( y = mx \). Understanding this, every line mentioned will pass through the origin.

Step 2: Calculate Line Equations for Part (a)

For each given slope, write the equation of the line passing through \((0,0)\). 1. For slope 1: \( y = 1x \rightarrow y = x \)2. For slope 0: \( y = 0x \rightarrow y = 0 \) (This is the x-axis.)3. For slope \( \frac{1}{2} \): \( y = \frac{1}{2}x \)4. For slope 2: \( y = 2x \)5. For slope -1: \( y = -x \)

Step 3: Sketch Lines for Part (a)

Plot each line on a graph.1. \( y = x \): Diagonal line through origin at a 45-degree angle.2. \( y = 0 \): Horizontal line along the x-axis.3. \( y = \frac{1}{2}x \): Line through the origin with slope shallower than \( y = x \).4. \( y = 2x \): Steeper diagonal line through the origin.5. \( y = -x \): Diagonal line through origin mirrored along the x-axis.

Step 4: Calculate Line Equations for Part (b)

Write the equation of each line passing through \((0,0)\) for slopes given in part (b).1. For slope \( \frac{1}{3} \): \( y = \frac{1}{3}x \)2. For slope \( \frac{1}{2} \): \( y = \frac{1}{2}x \)3. For slope \( -\frac{1}{3} \): \( y = -\frac{1}{3}x \)4. For slope 3: \( y = 3x \)

Step 5: Sketch Lines for Part (b)

Plot each line on a graph.1. \( y = \frac{1}{3}x \): Line through origin, shallower slope than \( y = x \).2. \( y = \frac{1}{2}x \): Matches a line from part (a), similar procedure.3. \( y = -\frac{1}{3}x \): Shallow negative slope through origin.4. \( y = 3x \): Steep positive line through origin.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a popular way to express linear equations. It is represented as \( y = mx + c \), where \( m \) denotes the slope of the line, and \( c \) represents the y-intercept, which is the point where the line crosses the y-axis.
When a line passes through the origin, the y-intercept \( c \) equals zero. Therefore, the equation simplifies to \( y = mx \). This form is quite useful for easily sketching and understanding graphs.
Here's why:

• The slope \( m \) indicates the steepness of the line and the direction in which it tilts. If \( m \) is positive, the line slopes upwards as you move from left to right.
• If \( m \) is negative, the line slopes downwards. A slope of zero results in a horizontal line.
• By knowing the slope, you can predict how the line behaves across any two points on a graph.

Mastering the slope-intercept form is key to understanding and solving many linear algebra problems.

Line Equations

Line equations are mathematical expressions that describe straight lines on a graph. With the slope-intercept form \( y = mx + c \), the key focus is on the values of \( m \) and \( c \).
Given the slope \( m \), these equations can inform us about the path of the line. For example, let's consider the slopes provided in the exercise, such as 1, 0, \( \frac{1}{2} \), and -1.

• For \( m = 1 \), the line equation becomes \( y = x \), indicating a diagonal line that increases steadily.
• For \( m = 0 \), the equation \( y = 0 \) stands for a horizontal line, also known as the x-axis.
• For \( m = \frac{1}{2} \), \( y = \frac{1}{2}x \), we observe a line that rises slower compared to \( y = x \).
• With \( m = -1 \), \( y = -x \) forms a line that decreases to the right, mirroring that of \( y = x \).

The equation forms the backbone of any graphing task and helps visualize the behavior of linear relationships.

Graphing Linear Equations

Graphing linear equations is a fundamental skill in understanding algebra. Once you have your line in the form of \( y = mx + c \), placing it on a coordinate plane gives a visual representation of the mathematical relationship.
For any equation, you only need two points to draw the entire line, but with the usage of slope-intercept form, you can derive the entire behavior with just the slope and the y-intercept.

• Start by plotting the y-intercept (for example, in lines through the origin, this means starting at \((0,0)\)).
• Use the slope \( m \) to determine the next point. For instance, a slope of 1 means for every step right, take a step up.
• Connect these points with a ruler to extend your line across the graph.

Practicing the graphing of linear equations enables you to visually interpret solutions and better understand the abstract representations given by equations.