Problem 47 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The slope of the line is -1/2 and the y-intercept is 0. The graph of the function is a line that passes through the origin, going down from left to right due to a negative slope.

Step 1: Identify the Slope

The slope (m) of a line is the coefficient of the x in the equation. In g(x) = -1/2x, the slope is -1/2. This is the rate of change of the function, indicating that for every increase of 1 in x, y decreases by 1/2.

Step 2: Identify the y-intercept

The y-intercept (b) is the constant term in the linear equation, indicating where the line crosses the y-axis. In this equation, there is no constant term, which means that the y-intercept is 0. This indicates that the line passes through the origin (0,0).

Step 3: Graph the Function

Start by marking the y-intercept (0,0) on the graph. Because the slope is -1/2, for every 2 units move to the right along the x-axis (increase in x), move 1 unit down along the y-axis (decrease in y), as the slope is negative. Plot these points and draw a line through them to represent the function g(x) = -1/2x.

Key Concepts

Slope of a Line

Exploring the concept of the slope is like unraveling the 'steepness' of a hill; it tells us how sharply a line tilts. In mathematics, the slope is denoted by the letter 'm' and mathematically, it represents the ratio of the rise (vertical change) to the run (horizontal change) as you move from one point to another along the line. To find the slope, you can use the formula \( m = \frac{\text{rise}}{\text{run}} \).

For the given exercise, the slope is -1/2, extracted directly from the linear function \( g(x) = -\frac{1}{2} x \). This tells us that the line moves down half a unit for every one unit it moves to the right. The negative sign indicates the line slopes downward as we move from left to right. Visualizing this can be a great help; picture walking down a gentle hill, descending half a meter for every meter you stroll forward.

Y-Intercept

The y-intercept is the point where the line intersects the y-axis. It is often associated with the value of 'b' in the slope-intercept form of a linear equation, which is \( y = mx + b \). This coordinate tells us where to start plotting our line on a graph.

In our example, there is no explicit 'b' term in the equation \( g(x) = -\frac{1}{2} x \), so by default, the y-intercept is 0. This means that the line crosses the y-axis right at the origin, where \( x = 0 \). It's like starting a journey from the very center of a map. With a y-intercept of zero, it simplifies plotting since we begin at the origin before using the slope to determine the direction and steepness of the line.

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations form straight lines when graphed and can be represented in various forms, including slope-intercept form, point-slope form, and standard form.

The most common is the slope-intercept form, \( y = mx + b \), where 'm' is the slope and 'b' is the y-intercept discussed earlier. This form makes graphing straightforward. The equation \( g(x) = -\frac{1}{2} x \) is an example of a linear equation, showing a direct, proportional relationship between 'x' and 'y'. The 'x' represents the independent variable, while 'g(x)' or 'y' is the dependent variable that changes in response to 'x'.

Linear equations are foundational for understanding much more complex functions in algebra and they are everywhere around us—describing everything from the rate of a car to the pitch of a roof.