Problem 13 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-14: Sketch the graph of \(y=m x\) for the given values of \(m\). $$ m=3,-2, \frac{2}{3},-\frac{1}{4} $$

Step-by-Step Solution

Answer

Sketch lines through the origin with slopes 3, -2, 2/3, -1/4.

Step 1: Understand the equation

The equation given is a linear equation: \( y = mx \), where \( m \) represents the slope of the line. Our objective is to sketch the graph for different values of \( m \).

Step 2: Plot the graph for m=3

For \( m = 3 \), the line equation is \( y = 3x \). This means that for every unit increase in \( x \), \( y \) increases by 3 units. The line passes through the origin (0,0) and has a slope of 3. It will be a steep line rising to the right.

Step 3: Plot the graph for m=-2

For \( m = -2 \), the line equation is \( y = -2x \). Here, for every unit increase in \( x \), \( y \) decreases by 2 units. The line passes through the origin and has a slope of -2. It will be a steep line falling to the right.

Step 4: Plot the graph for m=2/3

For \( m = \frac{2}{3} \), the line equation is \( y = \frac{2}{3}x \). This means that for every increase of 3 units in \( x \), \( y \) increases by 2 units. The line passes through the origin and is less steep compared to previous cases.

Step 5: Plot the graph for m=-1/4

For \( m = -\frac{1}{4} \), the line equation is \( y = -\frac{1}{4}x \). For every increase of 4 units in \( x \), \( y \) decreases by 1 unit. The line passes through the origin and has a mild slope, with a gentle decline to the right.

Step 6: Compare and understand the slopes

Notice that positive values of \( m \) result in lines that rise as \( x \) increases, whereas negative values of \( m \) result in lines that fall. The absolute value of \( m \) determines the steepness of the line.

Key Concepts

Slope of a Line

In the context of linear equations, the slope is a crucial concept. The slope, represented by the letter m in the equation \( y = mx \), indicates how steep a line is. Essentially, it tells us how much \( y \) changes for each unit change in \( x \).
When \( m \) is positive, the line rises as you move from left to right. Conversely, if \( m \) is negative, the line falls as you move in the same direction. The slope is expressed as a ratio, like \( \frac{rise}{run} \), which compares the vertical change (rise) to the horizontal change (run).

• A steeper line means a larger absolute value of the slope.
• A slope of zero results in a horizontal line.
• An undefined slope occurs for vertical lines.

Understanding the slope helps in predicting the direction and steepness of the line, which is pivotal when sketching a graph.

Graph Plotting

Graph plotting involves drawing a line on the graph based on a given equation. In this exercise, the equation is \( y = mx \), a linear equation that passes through the origin (0,0). To plot such a graph, you need to determine how the line behaves based on the slope \( m \).
Let's go through the steps:

• Find the origin (0,0) on the graph; this is the point where all lines from \( y = mx \) equations pass through.
• Use the slope \( m \) to find another point. For example, with \( m = 3 \), starting from the origin, move up 3 units and right 1 unit to find another point on the graph.
• Draw the line using these two points.

This method works for any value of \( m \), whether it is positive, negative, or a fraction. Practice plotting helps you visualize the slope and reinforce your understanding of linear equations.

Linear Graph Characteristics

Understanding the characteristics of a linear graph is essential for analyzing and interpreting equations. These graphs have distinct features worth noting:

• Every line in the form of \( y = mx \) is straight and passes through the origin. This makes them easy to identify.
• The slope \( m \) dictates the tilt and direction of the line. High positive slopes make the line steep and rising, while high negative slopes make it steep and falling.
• A smaller absolute value of \( m \) results in a flatter line, such as the line with \( m = -\frac{1}{4} \), which declines gently.

Linear graphs help in visualizing relationships between variables and can be used for making predictions based on linear trends. Recognizing these characteristics allows better understanding and manipulation of linear equations in mathematical problems.