Problem 70 - College Algebra

Question

An \(x\) -intercept of \((-4,0)\) and \(y\) -intercept of \((0,-2)\)

Step-by-Step Solution

Answer

The equation of the line is \( y = -\frac{1}{2}x - 2 \).

Step 1: Identify Intercepts

To find the equation of a line given the intercepts, first identify the intercepts. Here, the x-intercept is (-4,0) and the y-intercept is (0,-2).

Step 2: Determine the Slope

The slope (m) of a line is given by \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Plugging the intercepts into this formula, we have \( m = \frac{-2 - 0}{0 - (-4)} = \frac{-2}{4} = -\frac{1}{2} \).

Step 3: Write the Equation in Point-Slope Form

Using one of the intercepts and the slope, we can write the equation in point-slope form: \( y - y_1 = m(x - x_1) \). Using (0, -2) as the point, the equation is \( y + 2 = -\frac{1}{2}x \).

Step 4: Convert to Slope-Intercept Form

To convert the equation to the slope-intercept form \( y = mx + b \), solve for y: \( y = -\frac{1}{2}x - 2 \). This is the equation of the line with the given intercepts.

Key Concepts

Understanding the X-Intercept

An x-intercept is where a graph crosses the x-axis. This means that at this point, the value of y is zero. In the ordered pair representing the x-intercept, the y-component is always zero.

• For example, in the ordered pair \((-4, 0)\), the graph intersects the x-axis at -4.
• An intercept on the x-axis provides a point through which the graph passes. This information is useful when plotting the graph.
• It can easily help in calculating the slope if at least one more point of the line is provided, such as the y-intercept.

Exploring the Y-Intercept

The y-intercept is a point where the line crosses the y-axis, which implies that the x-coordinate is zero at this point. In an ordered pair, the x-component is always zero for a y-intercept. When given as \((0, -2)\), this means the line crosses the y-axis at -2.

• The y-intercept can be used to conveniently start graphing a line on a coordinate plane.
• This point also allows easy transformation of linear equations into the standard forms we often use, such as slope-intercept form.
• In any linear equation of the form \(y = mx + b\), the y-intercept is represented by \(b\).

What is Slope?

The slope of a line is a measure of how steep the line is. It reflects the vertical change for each unit of horizontal change. This ratio is crucial in defining the rise over the run between two points on a line, represented mathematically as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

• In our example, using points \((-4, 0)\) and \((0, -2)\), the slope is calculated as \(-\frac{1}{2}\).
• A positive slope means the line inclines upwards as you move from left to right.
• A negative slope, like in this case, indicates the line declines as you move from left to right.
• The slope tells us how much y increases or decreases as x increases.

Point-Slope Form Explained

Point-slope form allows us to describe a line given a point on the line and the slope. The formula is \( y - y_1 = m(x - x_1) \). This form is especially useful when you do not have information about the y-intercept.

• Using the y-intercept \((0, -2)\) and the slope \(-\frac{1}{2}\), the equation becomes \(y + 2 = -\frac{1}{2}x\).
• It's convenient because a specific point (x_1, y_1) on the line and the slope \(m\) are integrated into the equation directly.
• This form can be easily adjusted into other linear forms, such as slope-intercept form.

Simplifying with Slope-Intercept Form

Slope-intercept form is one of the most straightforward ways to express a line's equation, written as \( y = mx + b \). The key components include the slope \(m\) and the y-intercept \(b\).

• For our line, the equation converts to \(y = -\frac{1}{2}x - 2\).
• This form makes it simple to quickly identify the y-intercept \(-2\) directly from the equation.
• Once known, it allows for easy sketching of the line on a graph since the slope and y-intercept are immediately identifiable.
• This form is prevalent in algebra because of its straightforward nature and ease of application in graphing.