Problem 7 - A Graphical Approach to Precalculus with Limits

Question

Write the slope-intercept form of the line that passes through the given point with slope \(m .\) $$\text { Through }\left(\frac{1}{2},-4\right), m=2$$

Step-by-Step Solution

Answer

The slope-intercept form is \( y = 2x - 5 \).

Step 1: Recall the slope-intercept form equation

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

Step 2: Substitute the slope into the equation

Given the slope \( m = 2 \), substitute this value into the slope-intercept form equation. The equation becomes \( y = 2x + b \).

Step 3: Plug the point into the equation to solve for b

The line passes through the point \( \left( \frac{1}{2}, -4 \right) \). Substitute \( x = \frac{1}{2} \) and \( y = -4 \) into the current equation \( y = 2x + b \):\[ -4 = 2\left(\frac{1}{2}\right) + b \]

Step 4: Simplify and solve for b

Simplify \( 2 \times \frac{1}{2} \) to get 1, so the equation becomes:\[ -4 = 1 + b \]Subtract 1 from both sides to find \( b \):\[ b = -4 - 1 = -5 \]

Step 5: Write the final equation

Now that we have the value for \( b \), substitute \( b = -5 \) back into the slope-intercept form equation. Therefore, the equation of the line is:\[ y = 2x - 5 \]

Key Concepts

Linear Equations

Linear equations are mathematical expressions that create a straight line when graphed on a coordinate plane. The simplest form of a linear equation is the slope-intercept form, which is often written as \( y = mx + b \). Here, \( m \) indicates the slope, and \( b \) represents the y-intercept.

Linear equations show the relationship between two variables, typically \( x \) and \( y \). Each point on the graph corresponds to an \( (x, y) \) pair that satisfies the equation.

• For example, \( y = 2x - 5 \) is a linear equation with:
  • Slope \( m = 2 \)
  • Y-intercept \( b = -5 \)

This tells us how the value of \( y \) changes in relation to \( x \). In a practical sense, linear equations help us model relationships in real-world scenarios, such as predicting costs or calculating distances.

Slope

The slope of a line in linear equations is a crucial element that defines the line’s steepness and direction. It is denoted by \( m \) in the slope-intercept form of the linear equation \( y = mx + b \).

The slope is calculated as "rise over run". This means it is the ratio of the vertical change to the horizontal change between two points on the line. For example:

• If the slope \( m = 2 \), as in our example equation, it indicates that for a unit increase in \( x \), \( y \) increases by 2 units.

Different types of slopes include:

• Positive slope: The line ascends to the right.
• Negative slope: The line descends to the right.
• Zero slope: A horizontal line.
• Undefined slope: A vertical line.

The ability to determine the slope is essential in understanding the direction and exact nature of linear relationships.

Y-Intercept

In the context of a linear equation, the y-intercept is the point where the line crosses the y-axis. It is represented by \( b \) in the equation \( y = mx + b \). This value tells us where the line intersects when \( x \) is zero.

For example, in the equation \( y = 2x - 5 \), the y-intercept is \( -5 \). This means when \( x = 0 \), the value of \( y \) is \(-5\). The line touches the y-axis at the point \( (0, -5) \).

• The y-intercept provides a starting point for graphing the equation.
• It is essential for configuring the equation in its proper form.

By combining knowledge of both the slope and y-intercept, you can effectively graph a line or analyze its behavior on a graph, providing valuable insights into the underlying relationship between the variables involved.