Problem 72 - College Algebra

Question

For the following exercises, sketch a line with the given features. A \(y\) -intercept of (0,7) and slope \(-\frac{3}{2}\)

Step-by-Step Solution

Answer

Plot (0,7), use slope \(-\frac{3}{2}\) to find (2,4), and draw the line.

Step 1: Understanding the y-intercept

The y-intercept is the point where the line crosses the y-axis. In this exercise, the y-intercept is given as (0,7), so the line will pass through the point (0,7) on the graph.

Step 2: Identifying the Slope

The slope of a line describes how steep the line is. A slope of \(-\frac{3}{2}\) means that for every 2 units moved to the right (positive direction on x-axis), the line moves 3 units down (negative direction on y-axis). The negative sign indicates a downward slope as you move from left to right.

Step 3: Plotting the y-intercept

To begin, plot the point (0,7) on a graph. This point is on the y-axis, directly 7 units above the origin (0,0).

Step 4: Using the Slope to Find Another Point

Starting from the y-intercept (0,7), use the slope to find another point on the line. From (0,7), move 2 units to the right to x=2, then move 3 units down to y=4, landing at the point (2,4).

Step 5: Drawing the Line

With the points (0,7) and (2,4) plotted, draw a straight line through these points. This line accurately represents a line with a y-intercept of (0,7) and a slope of \(-\frac{3}{2}\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way to write the equation of a line so that it's simple to graph and understand. This form is written as \(y = mx + b\), where \(m\) represents the slope and \(b\) is the y-intercept. Using the slope-intercept form allows us to quickly see how a line behaves by observing its slope and where it crosses the y-axis. This form is especially useful as it gives two critical pieces of information at a glance:

• The slope \(m\), which tells us the line's steepness and direction.
• The y-intercept \(b\), which is the starting point on the y-axis.

Applying this to our exercise, the equation of the line, given a y-intercept of 7 and a slope of \(-\frac{3}{2}\), would be \(y = -\frac{3}{2}x + 7\). This indicates that the line decreases at a rate of \(-\frac{3}{2}\) as it moves along the x-axis.

Graphing Lines

Graphing lines involves plotting points on a coordinate plane based on a given equation or set of features. To accurately graph a line:

• Start by marking the y-intercept on the y-axis. This is the point where the line will cross the axis.
• Use the slope to determine the direction and steepness of the line. From the y-intercept, move according to the slope to find the next point.
• Draw a straight line through these points to see the full line.

In our exercise, after plotting the y-intercept at (0,7), we use the slope \(-\frac{3}{2}\) to move 2 units to the right and 3 units down to locate another point on the line at (2,4). Graphing involves accurately connecting these points and extending the line in both directions.

Y-Intercept

The y-intercept is a critical feature when analyzing or graphing a line. It is the point where the line crosses the y-axis, and it tells you the value of \(y\) when \(x = 0\). In the context of the equation \(y = mx + b\), the y-intercept is always \(b\). Knowing the y-intercept helps us start plotting the line on the graph.In our exercise, the y-intercept is (0,7). This means that any values of \(x\) will start from this point when plotted, making it essential for accurate line drawing. The y-intercept provides a solid starting foundation, allowing us to apply the slope effectively from this point.

Slope

The slope of a line quantifies its steepness and direction. It is represented as a ratio of the "rise" over the "run," indicating the amount of vertical change for every unit of horizontal movement. In general, the slope \(m\) can be calculated using two points on a line: \((x_1, y_1)\) and \((x_2, y_2)\), using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\).In our exercise, we use a slope of \(-\frac{3}{2}\). This means that for every 2 units moved horizontally to the right along the x-axis, the line will move 3 units vertically downward. A negative slope indicates that the line descends as it moves from left to right:

• Positive slope: Line moves upward.
• Negative slope: Line moves downward.
• Zero slope: Horizontal line.
• Undefined slope: Vertical line.

Understanding the slope is essential in determining how to position and draw the line correctly on a graph.