Problem 12 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the lines through the point with the indicated slopes on the same set of coordinate axes. Point (-4,1) Slopes (a) 4 \((b)-2\) (c) \(\frac{1}{2}\) (d) Undefined

Step-by-Step Solution

Answer

The sketch should have a common point (-4,1) for all lines. Line (a) should be moving upward to the right, line (b) should be moving downward to the right, line (c) should be moving slightly upward to the right and line (d) should be a vertical line.

Step 1: Sketching Line (a)

Start by plotting the point (-4,1). This is a common point for all 4 lines. A line with a slope of 4 means that for every 1 unit the x-coordinate moves to the right, the y-coordinate moves up by 4 units. Move to the right to plot several points and then draw the line passing through these points, making sure it also passes through the point (-4,1).

Step 2: Sketching Line (b)

A line with a slope of -2 means that for every 1 unit the x-coordinate moves to the right, the y-coordinate moves down by 2 units. Using the point (-4,1), move to the right to plot several points and then draw the line passing through these points.

Step 3: Sketching Line (c)

A line with a slope of 1/2 means that for every 2 units the x-coordinate moves to the right, the y-coordinate moves up by 1 unit. Using the point (-4,1), move to the right to plot several points and then draw the line passing through these points.

Step 4: Sketching Line (d)

An undefined slope leads to a vertical line. Therefore, draw a vertical line passing through the point (-4,1). This line does not have any slope and is perpendicular to the x-axis.

Key Concepts

Slope of a Line

The slope of a line is a fundamental concept in coordinate geometry. It indicates the direction and steepness of a line on a graph. You can think of it as a measure of how much the line "rises" or "falls" as it moves along the x-axis. The mathematical formula for the slope, often represented as "m," is calculated as the change in the y-coordinate divided by the change in the x-coordinate, or \( m = \frac{\Delta y}{\Delta x} \).

• A positive slope means the line ascends as it goes from left to right.
• A negative slope means the line descends as it moves from left to right.
• A larger absolute value of the slope indicates a steeper line.

Understanding slope is crucial because it not only tells us how to draw a line but also helps in understanding its behavior. For example, with a slope of 4, for each step you take to the right, you move up 4 steps on the y-axis.

Sketching Lines

Sketching lines on a coordinate plane involves plotting points and using the slopeto guide the line's direction. Start by plotting a given point, such as (-4,1), on the graph. This point serves as a fixed reference. Next, using the slope, plot subsequent points.For example:

• With a slope of 4, begin at (-4,1) and move 1 unit right and 4 units up repeatedly to plot multiple points. Connect them to form the line.
• For a slope of -2, start again at (-4,1), move 1 unit right and 2 units down.
• With slope \( \frac{1}{2} \), for every 2 units you move right, go up 1 unit.

The key is consistency. Always use the same rise-over-run method to maintain the correct angle of the line. Connecting these points will give a clear visual representation of the line on the coordinate plane, demonstrating how the slope affects its steepness and direction.

Point-Slope Form

Point-slope form is an equation format that allows you to write the equation of a line when you know one point on the line and the slope. The formula is expressed as \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is a point on the line, and \(m\) is the slope.This form is particularly useful because it provides a straightforward method to develop the line's equation without needing additional calculations like finding the y-intercept first. It emphasizes the focus on slope and the given point. For instance, if you have a point (-4, 1) and a slope 4, your equation would be \( y - 1 = 4(x + 4) \).Using point-slope form simplifies the process of defining a line and is a handy tool for sketching and understanding line behavior in coordinate geometry.

Vertical Lines

Vertical lines are unique in coordinate geometry because they have an undefined slope. Thisundefined nature arises because vertical lines have no "run" or change in the x-coordinate, which leads to a division by zero situation when calculating slope.A vertical line is a line that goes straight up and down. It is perpendicular to the x-axis and is represented by an equation of the form \( x = a \), where \(a\) is the constant x-value for all points on the line. In the context of our original exercise, a vertical line through the point (-4,1) would simply be \( x = -4 \).Understanding vertical lines is crucial because they defy the typical behavior of other lines regarding slopes, leading to unique equation formats in coordinate geometry.