Problem 72 - College Algebra with Corequisite Support

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 to reach (2,4), and draw the line.

Step 1: Identify the y-intercept

The given y-intercept is (0,7). This means the line crosses the y-axis at the point (0,7). This is a starting point for graphing the line.

Step 2: Understand the Slope

The slope of the line is given as \(-\frac{3}{2}\). This means for every 2 units the line moves horizontally to the right (positive direction), it moves 3 units vertically down (negative direction).

Step 3: Plot the y-intercept

On a coordinate graph, locate the y-axis and mark the point (0,7). This is where the line will cross the y-axis.

Step 4: Apply the Slope from y-intercept

Starting at the point (0,7), use the slope \(-\frac{3}{2}\) to find another point on the line. Since the slope is \(-\frac{3}{2}\), move 2 units to the right (positive x-direction) and 3 units down (negative y-direction) to reach the point (2,4).

Step 5: Draw the Line

Using a ruler, draw a straight line through the points (0,7) and (2,4). This line represents the equation given with the y-intercept (0,7) and slope \(-\frac{3}{2}\).

Key Concepts

y-intercept

The **y-intercept** is a critical concept in graphing linear equations. It is the point where the line crosses the y-axis, and it can be identified by the coordinates

• (0, b)

The "0" signifies that when x is 0, the line's position is only determined by the y-value, represented here as "b." For our specific example, the y-intercept is (0,7). This means the line crosses the y-axis directly at the point (0,7), hence we start plotting our line from this very point. Remember, if you were to replace "b" with another number, the intersection on the y-axis would change accordingly.

slope

Let's delve into the concept of **slope**, an essential characteristic of a line that indicates its direction and steepness. In the exercise, the slope is

• -\(\frac{3}{2}\)

This slope tells us that for every 2 units we move horizontally to the right, we move 3 units down vertically. The formula for slope is generally depicted as "rise over run," or:\[m = \frac{\Delta y}{\Delta x} = \frac{\text{change in y}}{\text{change in x}} \]
A negative slope, like \(-\frac{3}{2}\), indicates that the line will tilt downwards as it moves from left to right, signifying a downward trend. If the slope was positive, we would move upward instead.

coordinate graph

A **coordinate graph** is essentially a two-dimensional number line used to demonstrate various mathematical equations visually. This grid comprises two axes:

• The horizontal line is known as the x-axis.
• The vertical line is called the y-axis.

They intersect at a central point called the origin, designated as (0,0). Each point on this graph is expressed as a pair of numbers usually represented as (x, y). In our scenario, we use a coordinate graph to map our points like (0,7) and (2,4) to draw a line that portrays our equation. Understanding a coordinate graph is vital as it provides the visual framework for plotting and interpreting points, lines, and curves.

plotting points

**Plotting points** is an essential skill for graphing equations. In our exercise, this means taking particular points and placing them accurately on the coordinate graph. After specifying the y-intercept at (0,7), you need to start plotting here. Next, utilize the slope of \(-\frac{3}{2}\) to determine another point. From (0,7), move 2 units to the right, then go down by 3 units. This lands you at the new point (2,4). By marking these points on the graph, you create a visual guide that outlines where the linear path will go. The final step is connecting these dots with a straight edge, forming a line that accurately represents the given linear equation.