Problem 8 - Algebra 1: Concepts and Skills

Question

find the slope and y-intercept of the equation. $$y=\frac{1}{2} x+6$$

Step-by-Step Solution

Answer

The slope of the function is \(\frac{1}{2}\) and the y-intercept is \(6\).

Step 1: Identify the slope

In the slope-intercept form \( y = mx + c \), 'm' represents the slope of the function. Comparing this with the given equation \( y = \frac{1}{2}x + 6 \), we see that \( m = \frac{1}{2} \). This means the slope of the function is \( \frac{1}{2} \).

Step 2: Identify the y-intercept

In the slope-intercept form \( y = mx + c \), 'c' represents the y-intercept of the function. Comparing this with the given equation \( y = \frac{1}{2}x + 6 \), we see that \( c = 6 \). This means the y-intercept of the function is \( 6 \).

Key Concepts

Understanding the Y-intercept

In a linear equation, particularly in the slope-intercept form, the y-intercept is an essential element. It is the point where the line crosses the y-axis. This happens when the value of \( x \) is zero. So, it provides the starting value of \( y \) when \( x \) doesn't contribute any value. In the equation \( y = \frac{1}{2}x + 6 \), the y-intercept is \( 6 \). This means that when \( x = 0 \), the value of \( y \) will be \( 6 \).

• The y-intercept helps in plotting the graph of a line quickly, knowing where it begins on the vertical axis.
• A positive y-intercept indicates the line crosses the y-axis above the origin, while a negative one indicates it crosses below.

Demystifying the Slope-Intercept Form

The slope-intercept form of a linear equation is commonly expressed as \( y = mx + c \). Here, \( m \) represents the slope of the line, and \( c \) is the y-intercept. This form is extremely useful in algebra as it gives direct insights into the line's direction and initial position.

• The slope \( m \) indicates how steep a line is, showing how much \( y \) changes with a change in \( x \).
• The y-intercept \( c \) reveals where the line intersects the y-axis. This can be understood easily by substituting \( x = 0 \) in the equation.
• Using the slope-intercept form makes graphing lines simpler and allows for quick interpretation of both the slope and y-intercept from given functions.

Recognizing this form allows students to swiftly determine the characteristics of the line and predict its behavior over different values of \( x \). It's particularly beneficial in real-world problems to describe relationships between variables.

Delving into Linear Equations

Linear equations describe a straight line on a graph and are written in several forms, with the slope-intercept form being the most popular due to its simplicity. In these equations, every term is either a constant or the product of a constant and a single variable. Linear equations are characterized by having no exponents higher than one, making them easy to work with.

• They are fundamental in algebra and serve as a stepping stone to more complicated subjects such as calculus.
• They can represent various relationships or trends in fields like economics, physics, and biology.
• Understanding linear equations helps students solve problems involving rates of change and settings where two quantities are directly proportional.

In our example, the equation \( y = \frac{1}{2}x + 6 \) is a perfect demonstration of a linear equation. The ease with which you can predict outcomes or visualize growth patterns makes understanding linear equations a valuable skill in everyday life.