Problem 79 - College Algebra

Question

For the following exercises, sketch the graph of each equation. $$ k(x)=\frac{2}{3} x-3 $$

Step-by-Step Solution

Answer

Graph the line with slope \( \frac{2}{3} \) and y-intercept \( -3 \).

Step 1: Identify the Type of Equation

The given equation is in the form \( y = mx + b \), which represents a linear equation. Here, \( m \) is the slope, and \( b \) is the y-intercept. In the equation \( k(x) = \frac{2}{3}x - 3 \), the slope \( m \) is \( \frac{2}{3} \) and the y-intercept \( b \) is \( -3 \).

Step 2: Plot the Y-Intercept

Start by plotting the y-intercept on the coordinate plane. The y-intercept is the point where the line crosses the y-axis. For the equation \( k(x) = \frac{2}{3}x - 3 \), the y-intercept is \( (0, -3) \). Mark this point on the graph.

Step 3: Use the Slope to Find Another Point

The slope \( \frac{2}{3} \) means that for every 3 units you move to the right, you move up 2 units. Starting from the y-intercept \( (0, -3) \), move horizontally 3 units to the right to \( (3, -3) \) and then move 2 units up to \( (3, -1) \). Plot this point on the graph.

Step 4: Draw the Line Through the Points

With the points \( (0, -3) \) and \( (3, -1) \) plotted, draw a straight line through these points. Extend the line in both directions across the graph to indicate it continues indefinitely.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a simple and powerful way to express linear equations. It is written as \( y = mx + b \), where \( m \) represents the slope of the line, and \( b \) is the y-intercept. This form makes it easy to quickly identify both the slope and the y-intercept.

• The slope \( m \) tells you how steep the line is, or how much \( y \) changes for a given change in \( x \).
• The y-intercept \( b \) is the point where the line crosses the y-axis. It represents the value of \( y \) when \( x = 0 \).

Using the slope-intercept form makes graphing a linear equation straightforward, as you just need to identify these two components and apply them to the coordinate plane. This format also quickly communicates key information about the line's behavior in a mathematical context.

Graphing Linear Functions

Graphing linear functions involves plotting points on a coordinate plane and drawing a straight line through these points. Linear functions create straight lines, making them predictable and easy to work with.
To graph the function, start with key pieces of information: the slope and the y-intercept. Begin by:

• Plotting the y-intercept point on the graph.
• Using the slope to find other points on the line.

After plotting these points, draw a line through them to represent the linear function. Extend the line in both directions on the graph. This depiction shows how the function behaves across different values of \( x \). With practice, graphing linear functions becomes an efficient way to visualize equations.

Coordinate Plane

The coordinate plane is a two-dimensional surface where you can graph points, lines, and curves. It consists of two perpendicular axes: a horizontal axis (the x-axis) and a vertical axis (the y-axis).
Each point on the coordinate plane is represented by a pair of numbers \((x, y)\), which are the coordinates of the point:

• The x-coordinate indicates horizontal movement from the origin.
• The y-coordinate indicates vertical movement from the origin.

The origin \((0, 0)\) is the point where the x-axis and y-axis intersect. The coordinate plane is essential for graphing equations and visualizing mathematical concepts, allowing you to see the relationship between different variables clearly.

Y-Intercept

The y-intercept is a critical concept in understanding linear equations. It is the specific point on the graph where the line crosses the y-axis. This happens when the value of \( x \) is zero. Given the equation in slope-intercept form \( y = mx + b \), the y-intercept is simply \( b \).
For example, in the equation \( k(x) = \frac{2}{3}x - 3 \), the y-intercept is \(-3\), meaning the line will cross the y-axis at the point \((0, -3)\).Understanding the y-intercept helps in quickly sketching graphs, as it gives a definite starting point, from which you can use the slope to determine other points on the line. Recognizing where the line intersects the y-axis allows one to see the initial value of a function when plotted.