Problem 77 - College Algebra

Question

For the following exercises, sketch the graph of each equation. $$ g(x)=-3 x+2 $$

Step-by-Step Solution

Answer

The graph is a straight line with a slope of -3 and a y-intercept at (0, 2).

Step 1: Identify the Equation Type

The equation \(g(x) = -3x + 2\) is in the form of \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This is the equation of a straight line.

Step 2: Determine the Slope and Y-Intercept

For \(g(x) = -3x + 2\), the slope \(m\) is -3, and the y-intercept \(b\) is 2. This means the line crosses the y-axis at the point (0, 2).

Step 3: Plot the Y-Intercept

Start by plotting the y-intercept on the graph. Place a point on the y-axis at (0, 2).

Step 4: Use the Slope to Find Another Point

The slope of -3 means that for every 1 unit you move to the right (positive direction along the x-axis), you move 3 units down (since -3 is negative) along the y-axis. From (0, 2), move 1 unit right to (1, 2) and 3 units down to (1, -1). Plot the point (1, -1).

Step 5: Draw the Line

Connect the points (0, 2) and (1, -1) with a straight line. This line extends infinitely in both directions, following the slope of -3.

Key Concepts

Slope

When discussing the equation of a line, slope refers to how steep or flat the line is. The slope can be found in the general line equation:

• In the formula \(y = mx + b\), \(m\) stands for slope.
• A positive slope means the line rises as you move from left to right, while a negative slope means it descends.
• For a slope of -3, it moves 3 units down for each unit it goes to the right.

Understanding the slope helps you predict the direction and angle of your line on the coordinate plane.
It's a key concept in graphing because it gives you a method to create the line once you have an initial point.

Y-intercept

The y-intercept is where the line crosses the y-axis. In an equation like \(y = mx + b\), \(b\) is the y-intercept.

• This point is always found at \((0, b)\).
• For the equation \(g(x) = -3x + 2\), the y-intercept is 2, hence the line crosses the y-axis at the point \((0, 2)\).

Locating the y-intercept is crucial because it provides a starting point for drawing the line.
From this point, you can use the slope to determine how the line continues.

Equation of a Line

The equation of a line in slope-intercept form is represented as \(y = mx + b\). This form clearly shows both the slope and the y-intercept, making it simpler to plot a line.

• \(m\) stands for slope, dictating the line's rise over run.
• \(b\) is the y-intercept, telling us where the line meets the y-axis.

It's crucial to memorize this form because most linear equations you'll encounter are presented this way.
Having the equation in this form allows for quick graph sketching from just the slope and intercept values.

Coordinate Plane

The coordinate plane is a two-dimensional surface where you can plot equations and points. It consists of two axes:

• The horizontal axis is the x-axis.
• The vertical axis is the y-axis.

Each point on this plane is identified by a pair of numbers known as a coordinate, \((x, y)\).
When graphing an equation like \(g(x) = -3x + 2\), understanding how the coordinate plane works helps you place each calculated point precisely. It's the stage where your equations come to life, as the relationship between x and y is plotted as a visual line or curve.