Problem 117 - College Algebra with Modeling and Visualization

Question

Compare the slope-intercept form with the point-slope form. Give examples of each.

Step-by-Step Solution

Answer

Slope-intercept: \( y = mx + b \). Point-slope: \( y - y_1 = m(x - x_1) \).

Step 1: Understanding Slope-Intercept Form

The slope-intercept form of a linear equation is presented as \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept of the line. The slope, \( m \), tells us how steep the line is, while the y-intercept, \( b \), is the point where the line crosses the y-axis.

Step 2: Creating an Example of Slope-Intercept Form

Consider the line equation \( y = 2x + 3 \). In this example, the slope \( m \) is 2, indicating that for every unit increase in \( x \), \( y \) increases by 2 units. The y-intercept \( b \) is 3, meaning the line crosses the y-axis at \( y = 3 \).

Step 3: Understanding Point-Slope Form

The point-slope form of a linear equation is written as \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is a specific point on the line, and \( m \) is the slope. This form is useful when you know one point on the line and the slope.

Step 4: Creating an Example of Point-Slope Form

If a line passes through the point \((4, 5)\) with a slope \( m = 3 \), the point-slope form is written as \( y - 5 = 3(x - 4) \). This equation shows that the line goes through the point \((4, 5)\) and has a slope of 3.

Step 5: Comparing the Two Forms

The slope-intercept form \( y = mx + b \) is useful for quickly identifying and graphing the slope and y-intercept. In contrast, the point-slope form \( y - y_1 = m(x - x_1) \) is particularly helpful for writing an equation when a specific point and the slope are known. Ultimately, both forms convey the similar linear relationship but are used based on the specific information available or preferred.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is a straightforward and popular way to express a line. It is written as \( y = mx + b \). Here, \( m \) is the slope of the line, which indicates how the line rises or falls as you move along the x-axis. Meanwhile, \( b \) is the y-intercept, which is the point where the line crosses the y-axis. This makes it easy to quickly identify both the steepness of the line and where it starts on the graph.

**Example:**
Suppose you have the equation \( y = 2x + 3 \):

• The slope \( m = 2 \) tells us the line rises by 2 units for each 1 unit it moves to the right.
• The y-intercept \( b = 3 \) indicates the line crosses the y-axis at the point \( (0,3) \).

This form is useful for quickly assessing how the line behaves without the need for additional information about particular points.

Point-Slope Form

The point-slope form is another way to write the equation of a line and is particularly handy when you know one point on the line and the slope. The formula is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) represents a specific point on the line and \( m \) is the slope.

**Example:**
Imagine a line passing through the point \((4, 5)\) with a slope of 3. The point-slope equation would be \( y - 5 = 3(x - 4) \):

• The numerical value 3 shows how steep the line is, as it rises 3 units for every unit it moves horizontally.
• The point (4, 5) is clearly a part of the line, as per our equation.

This form is adaptable when you don't have the y-intercept ready but need to conceptualize the line graphically using a known point and slope.

Y-Intercept

The y-intercept is a special and simple concept in the study of linear equations. It is the y-coordinate of the point where the line intersects the y-axis. In the context of the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \).

**Reason for Importance:**

• The y-intercept provides a starting point to draw the line on a graph.
• It's often the easiest feature to determine directly from linear data.

When you have an equation like \( y = 2x + 3 \), reading off the y-intercept as 3 is immediate. This means if you set \( x = 0 \), then \( y = 3 \), which gives a clear way to plot where the line begins as it moves across the graph. Understanding the y-intercept, therefore, provides an accessible entry into analyzing how a line behaves visually.