Problem 47 - College Algebra

Question

Give the slope and \(y\) -intercept of each line Whose equation is given. Then graph the r function. \(g(x)--\frac{1}{2} x\)

Step-by-Step Solution

Answer

The slope of the line is \(m = -\frac{1}{2}\) and the y-intercept is at the origin \(c = 0\). Graphing this function results in a line that passes through the origin and slopes downwards to the right.

Step 1: Identify the slope

First, identify the slope of the line. The slope of the line is given by the coefficient in front of \(x\) in the equation of the line. So, in the function \(g(x) = -\frac{1}{2} x\), the slope \(m = -\frac{1}{2}\).

Step 2: Identify the y-intercept

Next, identify the y-intercept of the line. The y-intercept is the constant term in the equation, but in this case, the equation does not have a constant term. This means that the y-intercept is at the origin, which is at (0,0). So, \(c = 0\).

Step 3: Graph the function

Now, graph the function. First, plot the y-intercept (0,0). Then, use the slope to determine the direction of the line. The slope is negative, which means the line will be going downwards. Starting from the y-intercept, move down 1 unit for every 2 units moved to the right along the x-axis to draw the line. Because the slope is -1/2, for every 2 units we move to the right, we move down 1.

Key Concepts

Slope of a line

The slope of a line is a measure indicating how steep the line is. Think of the slope as the line's slant or inclination. It tells you how much the line goes up or down as you move left to right across the graph.
If the slope is positive, the line inclines upwards as it moves from left to right. Conversely, if the slope is negative, it slopes downwards as it moves from left to right.
In the equation of a line, the slope is represented by the letter \( m \) in the linear equation format \( y = mx + b \). In the given exercise, the equation is \( g(x) = -\frac{1}{2}x \), which means the slope \( m = -\frac{1}{2} \).

• A slope of \( -\frac{1}{2} \) implies that for every 2 units you travel horizontally to the right, you move down 1 unit vertically.

Understanding the slope is crucial because it not only reveals the direction in which the line extends but also hints at the steepness of the line.

y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. It is a vital concept as it provides a starting point on the graph from which you can begin plotting your line.
The y-intercept is represented by the letter \( b \) in the linear form \( y = mx + b \). This value is the point where \( x = 0 \).
In our example equation \( g(x) = -\frac{1}{2}x \), there is no constant term added to the expression, so the y-intercept \( b = 0 \).

• This means the line passes through the origin at the coordinate (0,0).

This intercept is significant because it helps set the starting point for drawing the line on a graph, making it easier to see how the line behaves as the x-value changes.

Plotting points on a graph

Plotting points on a graph is a way to visually represent the relationship between two variables, like \( x \) and \( y \) in a linear equation. It involves identifying specific points by using their coordinates and marking these points on the coordinate plane.
First, you'll start with the y-intercept. In the case of our function \( g(x) = -\frac{1}{2}x \), the y-intercept is at point (0,0).
From this point, use the slope to plot additional points:

• A slope of -1/2 from (0,0) means moving right 2 units and down 1 unit to reach the next point.
• Continue this pattern to find more points that lie on the line.

The plotted points form a straight line, also known as the graph of the equation.
Practice plotting various functions to get comfortable with this method, as it is fundamental in understanding linear relationships and how they can be graphically represented.