Problem 22 - A Graphical Approach to Precalculus with Limits

Question

Find the slope-intercept form of the equation of the line satisfying the given conditions. Do not use a calculator. Through \((0,-8)\) and \((4,0)\)

Step-by-Step Solution

Answer

The equation of the line is \( y = 2x - 8 \).

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of the equation of a line is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.

Step 2: Calculate the Slope

The slope \( m \) of a line 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 \((0, -8)\) and \((4, 0)\). Calculate the slope as follows:\[m = \frac{0 - (-8)}{4 - 0} = \frac{8}{4} = 2\]

Step 3: Find the Y-Intercept

Using the point \((0, -8)\), since the x-coordinate is 0, this means \( b = -8 \). Thus, the y-intercept is -8.

Step 4: Write the Equation in Slope-Intercept Form

Now that we know both the slope \( m \) and the y-intercept \( b \), plug these values into the slope-intercept form equation: \[y = 2x - 8\]

Key Concepts

Slope Calculation

Calculating the slope is a fundamental step in understanding linear equations. The slope tells us how steep a line is or, in more practical terms, how much the line rises or falls as we move along it. To calculate the slope \( m \), we use the formula:- \( m = \frac{y_2 - y_1}{x_2 - x_1} \)where - \((x_1, y_1)\) and \((x_2, y_2)\) are two points on the line. In the given exercise, these points are \((0, -8)\) and \((4, 0)\). By substituting the coordinates into the slope formula, we find:- \( m = \frac{0 - (-8)}{4 - 0} = \frac{8}{4} = 2 \).This result means that the line rises 2 units for every 1 unit it moves to the right. Understanding slope is crucial, as it determines the direction and steepness of the line.

Y-Intercept

The y-intercept is where the line crosses the y-axis. Simply put, it's the value of \( y \) when \( x \) is 0. In the context of a slope-intercept equation, \( y = mx + b \), \( b \) represents this intercept.From the exercise, we have one of our points as \((0, -8)\). Since the x-coordinate is 0, this point gives us the y-intercept directly, making \( b = -8 \). This means our line crosses the y-axis at -8.

• The y-intercept helps us understand where the line will start in terms of the y-values.
• It is also essential in graphing the line on a coordinate plane, providing a starting point before using the slope to identify other points on the line.

Remember, knowing the y-intercept is just as important as the slope, as it completes the picture of a line's equation.

Linear Equations

Linear equations are equations of the first degree, which means they have the highest exponent of the variable as 1. They graph into straight lines. The most common form of a linear equation is the slope-intercept form, \( y = mx + b \).

• "\( y = mx + b \)" shows a direct relationship between \( x \) and \( y \), with \( m \) being the slope and \( b \) as the y-intercept.
• The equation is complete once both the slope and the y-intercept are known.

In the given problem, after calculating the slope (\( m = 2 \)) and identifying the y-intercept (\( b = -8 \)), we can write down the linear equation as \( y = 2x - 8 \). This linear equation succinctly describes how \( y \) changes with \( x \).By understanding linear equations, students gain insight into relationships between quantities and learn how to predict or analyze trends through simple mathematical models.