Problem 14 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 13-14: Sketch the graph of \(y=m x\) for the given values of \(m\). $$ m=5,-3, \frac{1}{2},-\frac{1}{3} $$

Step-by-Step Solution

Answer

Sketch four lines with varying slopes: two steep and two gentle, with both positive and negative directions.

Step 1: Understand the Equation

The equation given is in the form of a linear equation, \( y = mx \), where \( m \) is the slope of the line. The slope indicates the steepness and direction of the line on a graph.

Step 2: Identify the Slopes

The slopes provided are \( m = 5, -3, \frac{1}{2}, -\frac{1}{3} \). Each slope will result in a different line on the graph.

Step 3: Graph the Line for \( m = 5 \)

For \( m = 5 \), plot the line \( y = 5x \). This line passes through the origin (0,0) and rises steeply as \( x \) increases, because the slope is positive and greater than 1.

Step 4: Graph the Line for \( m = -3 \)

For \( m = -3 \), plot the line \( y = -3x \). This line also passes through the origin (0,0) but decreases as \( x \) increases, indicating a negative slope.

Step 5: Graph the Line for \( m = \frac{1}{2} \)

For \( m = \frac{1}{2} \), plot the line \( y = \frac{1}{2}x \). This line passes through the origin and increases as \( x \) increases, but more gently than \( y = 5x \), due to the slope being a fraction less than 1.

Step 6: Graph the Line for \( m = -\frac{1}{3} \)

For \( m = -\frac{1}{3} \), plot the line \( y = -\frac{1}{3}x \). This line passes through the origin and decreases as \( x \) increases, but more gently than \( y = -3x \), due to the slope being a negative fraction greater than -1.

Step 7: Analyze the Graphs

Compare the steepness and direction of each line plotted. The larger positive \( m \) (like 5) results in a steeper upward line, larger negative \( m \) (like -3) results in a steeper downward line, while fractional slopes like \( \frac{1}{2} \) and \( -\frac{1}{3} \) result in gentler slopes.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in linear equations, representing how much the line rises or falls as it moves from left to right. It's noted by the letter "m" in the equation \( y = mx \). Essentially, slope measures the steepness of a line.

• When the slope is positive, the line ascends, climbing up as you move along the graph.
• Conversely, if the slope is negative, the line descends, dropping down.
• The greater the value of m, the steeper the line.

Understanding slope is crucial, as it directly impacts the direction and angle of a line on a graph. It's calculated as a ratio of the vertical change (rise) to the horizontal change (run) between two points. For example, the slope \( m = 5 \) implies that for every unit the line moves to the right, it rises 5 units up.
The slope can also be a fraction, such as \( \frac{1}{2} \), indicating a gentler incline: for every 2 units moved horizontally, the line rises just 1 unit. Learning about slopes helps in predicting the trajectory of lines, valuable for graphing them accurately.

Graphing Linear Equations

When graphing linear equations like \( y = mx \), it's important to start at the right point and understand how to draw the slope. All lines for equations in the form \( y = mx \) will pass through the origin point (0,0), making the process straightforward.

• With a positive slope, draw the line upwards from left to right.
• For a negative slope, draw the line downwards, still left to right.

Here's how you might graph these four slope values:- **m = 5**: Start at (0,0). Move 1 unit right, 5 units up for the point (1,5). Connect the points with a line.- **m = -3**: Begin at (0,0). Move 1 unit right, 3 units down for (1,-3). Draw your line through these.- **m = \( \frac{1}{2} \)**: Start at (0,0). Move 2 units right, 1 unit up. This point is (2,1).- **m = \( -\frac{1}{3} \)**: Again, start at (0,0). Move 3 units right, 1 unit down to point (3,-1).Graphing allows you to visually understand the effects of different slopes, reinforcing their impact on line angles and directions.

Positive and Negative Slopes

The nature of slopes, whether they are positive or negative, is key in determining the direction a line will take on a graph. Positive slopes create lines that ascend in the direction moving from left to right.
The line rises as the x-values increase. For instance, an equation with \( m = 5 \) or \( m = \frac{1}{2} \) are examples of positive slopes.

• Positive integer slopes (e.g., 5) indicate a steep upward line.
• Fractional positive slopes (e.g., \( \frac{1}{2} \)) result in a gentler incline.

Negative slopes produce the opposite effect. Lines descend from left to right, indicating a drop as the x-values increase. Examples are \( m = -3 \) or \( m = -\frac{1}{3} \).

• Negative integer slopes (e.g., -3) create steep downward lines.
• Fractional negative slopes (e.g., \( -\frac{1}{3} \)) show a softer decline.

Understanding these slope types helps in predicting the movement pattern of lines. Such insights allow for better navigation through graphing exercises, enhancing comprehension of linear equations.