Problem 9 - Algebra 1: Concepts and Skills

Question

Plot the points and draw the line that passes through them. Without finding the slope, determine whether the slope is positive, negative, zero, or undefined. $$ (-4,-3) \text { and }(0,-3) $$

Step-by-Step Solution

Answer

The slope of the line that passes through (-4, -3) and (0, -3) is zero.

Step 1: Plotting the Points

The given points are (-4, -3) and (0, -3). Plot these points on the Cartesian plane. Point (-4,-3) will lie on the 4th quadrant, to the left of the y-axis, while point (0,-3) lie on the x-axis.

Step 2: Drawing the Line

Draw a straight line that passes through these two points.

Step 3: Determining the Slope

Look at the line that has been drawn. A positive slope rises left to right, a negative slope falls left to right, a zero slope is horizontal, and an undefined slope is vertical. The line connecting your points runs horizontally so the slope is zero

Key Concepts

Plotting Points on a Cartesian Plane

When plotting points, you start with a Cartesian plane which is divided into four quadrants by the x-axis (horizontal) and y-axis (vertical). Each point is represented by a pair of numbers, known as coordinates, which are written as \( (x, y) \). The first number, \( x \), is the horizontal position and the second number, \( y \), is the vertical position.

To plot a point, you start from the origin (the point where the x-axis and y-axis intersect, which is \( (0, 0) \) ), move horizontally to the x-coordinate, and then vertically to the y-coordinate. If the x-coordinate is negative, move left from the origin. If it's positive, move right. For the y-coordinate, move up if it's positive and down if it's negative.

• For the point \( (-4, -3) \), begin at the origin, move 4 units left and 3 units down.
• For the point \( (0, -3) \), you only move 3 units down from the origin, as the x-coordinate is zero.

These instructions will ensure that you accurately place points on the plane.

Positive and Negative Slopes

The slope of a line tells us how steep the line is and the direction it tilts. Think of slope as the line's 'slant'. A line with a positive slope rises as it goes from left to right, like climbing up a hill. Conversely, a line with a negative slope falls as it moves from left to right, similar to descending a hill.

Mathematically, the slope is calculated as the change in the y-coordinate divided by the change in the x-coordinate between two points on the line. If both coordinates increase (\( \Delta x > 0 \) and \( \Delta y > 0 \) ) or decrease (\( \Delta x < 0 \) and \( \Delta y < 0 \) ) together, the slope is positive. If one coordinate increases while the other decreases, the slope is negative.

Characteristics of Slope:

• Positive Slope: Moves upwards from left to right.
• Negative Slope: Moves downwards from left to right.

Zero and Undefined Slope

Not all lines slant. Some may be completely horizontal or vertical. A horizontal line has a slope of zero because there is no change in the y-coordinate as you move along the line (\( \Delta y = 0 \) ). It's like walking on flat ground; you're not moving up or down, just side to side. This kind of line runs parallel to the x-axis.

In contrast, a vertical line has an undefined slope. This is because the change in the x-coordinate is zero (\( \Delta x = 0 \) ), and since you cannot divide by zero, the slope is considered undefined. Imagining a vertical climb without any horizontal movement at all resembles this type of line, which runs parallel to the y-axis.

• Zero Slope: Horizontal line, no vertical change.
• Undefined Slope: Vertical line, no horizontal change.