Problem 37 - Precalculus Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. \(x\) -intercept \(=-\frac{1}{2}\) and \(y\) -intercept \(=4\)

Step-by-Step Solution

Answer

The point-slope form of the equation is \(y - 4 = 8x\), and the slope-intercept form is \(y = 8x + 4\).

Step 1: Find the slope

We'll start by determining the slope of the line. Remember that slope, often represented by \(m\), is \(\Delta y / \Delta x\), the change in \(y\) over the change in \(x\). Given the \(x\)-intercept as -1/2 and \(y\)-intercept as 4, the slope \(m\) can be calculated as \(m = (4 - 0) / (0 - (-1/2)) = 8\). Therefore, the slope of the line is 8.

Step 2: Write the point-slope form

We use the point-slope form equation \(y - y_1 = m(x - x_1)\). Given the \(y\)-intercept is 4, we'll use the point (0,4) in our equation to get the point-slope form: \(y - 4 = 8(x - 0)\), which simplifies to \(y - 4 = 8x\).

Step 3: Write the slope-intercept form

Now, we turn the point-slope form \(y - 4 = 8x\) into the slope-intercept form \(y = mx + c\). Make \(y\) the subject of the equation to get \(y = 8x + 4\). So, the slope-intercept form of the line is \(y = 8x + 4\).

Key Concepts

Understanding Point-Slope Form

The point-slope form is a way to characterize a line using its slope and a specific point on the line. The general formula for the point-slope form is given by:

• \( y - y_1 = m(x - x_1) \)

Here, \(m\) represents the slope, and \((x_1, y_1)\) is a particular point through which the line passes. This form is especially helpful when you already have a point and the slope of the line.
In our exercise, after finding the slope to be 8 using the x-intercept and y-intercept provided, we used the y-intercept point \((0, 4)\). This simplifies our equation to:

• \( y - 4 = 8(x - 0) \)

Simplified further, this becomes \( y - 4 = 8x \). In essence, the point-slope form ties together the information we have about individual points and slopes to represent the entire line.

Exploring Slope-Intercept Form

The slope-intercept form is one of the most recognizable ways to express a line's equation. It provides a clear view of both the slope and the y-intercept of the line. The formula for slope-intercept form is:

• \( y = mx + c \)

where \(m\) indicates the slope of the line, and \(c\) is the y-intercept where the line crosses the y-axis.
In the given problem, after establishing the slope and utilizing the y-intercept at 4, we convert our equation from the point-slope form \( y - 4 = 8x \) to the slope-intercept form, resulting in:

• \( y = 8x + 4 \)

This transformation involves solving the equation for \(y\) to make it clear and intuitive to identify the line's characteristics. This form allows for quick and easy plotting of the line on a graph and understanding of the line's behavior.

How to Calculate Slope

Calculating the slope of a line involves understanding how much the line rises or falls as you move along it. Slope is often denoted by \(m\) and is mathematically defined as the ratio of the vertical change (rise) to the horizontal change (run) between two points on the line:

• \( m = \frac{\Delta y}{\Delta x} \)

In the exercise, the slope was calculated between the x-intercept \((-1/2, 0)\) and the y-intercept \((0, 4)\), using:

• \( m = \frac{4 - 0}{0 - (-1/2)} = 8 \)

The slope of +8 indicates a steep upward trend, showing that for every unit increase in \(x\), \(y\) increases by 8 units. Slope is a fundamental aspect of defining the direction and steepness of a line, making it crucial for both graphing and algebraic manipulation.