Problem 70 - College Algebra

Question

For the following exercises, sketch a line with the given features. An \(x\) -intercept of (-4,0) and \(y\) -intercept of (0,-2)

Step-by-Step Solution

Answer

The line with an equation of \(y = -\frac{1}{2}x - 2\) intersects the \(x\)-axis at \(-4\) and the \(y\)-axis at \(-2\).

Step 1: Identify the intercepts

We are given two points that the line intersects with: the \(x\)-intercept at \((-4, 0)\) and the \(y\)-intercept at \((0, -2)\). This means the line crosses the \(x\)-axis at \(-4\) and the \(y\)-axis at \(-2\).

Step 2: Calculate the slope

The slope \(m\) of the line that passes through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, the points are \((-4, 0)\) and \((0, -2)\), so the slope is \(m = \frac{-2 - 0}{0 - (-4)} = \frac{-2}{4} = -\frac{1}{2}\).

Step 3: Write the equation of the line

Using the slope-intercept form \(y = mx + b\), where \(b\) is the y-intercept, we can write the equation of the line. We have \(m = -\frac{1}{2}\) and \(b = -2\), so the equation is \(y = -\frac{1}{2}x - 2\).

Step 4: Sketch the graph

To sketch the graph, plot the points \((-4, 0)\) and \((0, -2)\) on a coordinate plane. Draw a straight line through these points. The line should have a negative slope, slanting downwards from left to right.

Key Concepts

Understanding the x-intercept

The x-intercept is a crucial concept when dealing with linear equations and graphs. It is the point where the line intersects the x-axis. This means the y-coordinate at this point is always 0. In our exercise, the x-intercept is at \((-4,0)\). Here, the line meets the x-axis at the point \(x = -4\).
What does this mean visually on a graph?

• Imagine the x-axis as a horizontal line, and when a line touches this axis, the point of contact is the x-intercept.
• This tells us that anywhere along the x-axis, the y-value is zero.

The x-intercept is useful for understanding where the graph crosses horizontally and plays a key role in graphing linear equations.

Exploring the y-intercept

The y-intercept is another fundamental concept to grasp when working with linear equations. It is where the line crosses the y-axis, meaning the x-coordinate is 0 at this point. For our exercise, the y-intercept is given as \((0, -2)\).
How does this affect the graph?

• The y-intercept indicates the starting point of the line when plotting it on a graph.
• At this point, the value of \(x\) is zero, which is why it is critical in forming the equation of the line using the slope-intercept form.

Understanding the y-intercept helps in determining how high or low the line starts on the vertical axis.

Comprehending the slope-intercept form

The slope-intercept form is a straightforward way to express the equation of a line. It is written as \(y = mx + b\), where \(m\) stands for the slope and \(b\) represents the y-intercept. From our exercise, the slope is calculated to be \(-\frac{1}{2}\), and the y-intercept is \(-2\). This allows us to write our line's equation as \(y = -\frac{1}{2}x - 2\).
Why is this form useful?

• This form provides a quick and easy way to graph a line, as you directly get important information about the slope and starting point.
• The slope \(m\) helps in understanding the direction and steepness of the line, indicating how much the line rises or falls as it moves from left to right.
• The y-intercept \(b\) tells where the line begins on the y-axis, providing a starting point for the graph.

Mastering the slope-intercept form enriches your toolkit for solving and graphing linear equations effortlessly.