Question

Multiple Choice What is the \(y\)-intercept of the graph of the equation \(y=\frac{1}{3} x+2\) ? (Lesson \(4-6\) ) (A) 3 (B) \(\frac{1}{3}\) (c) \(-2\) (D) 2

Short answer

The y-intercept is 2, option D.

Step-by-step solution

Identify the y-intercept in the equation

The equation is in the slope-intercept form, which is given by the formula \(y = mx + c\) where \(m\) is the slope and \(c\) is the y-intercept. Identify \(c\) in the given equation \(y = \frac{1}{3}x + 2\).

Extract the y-intercept value

From the equation \(y = \frac{1}{3}x + 2\), the value of \(c\) is 2. This means the y-intercept of the graph is 2.

Choose the correct multiple choice answer

Refer to the options given: (A) 3, (B) \(\frac{1}{3}\), (C) -2, (D) 2. Match the calculated y-intercept with the provided options. Since the y-intercept is 2, the correct option is (D).

Key concepts

Slope-Intercept Form

In algebra, the equation of a line is often expressed in the slope-intercept form, which is written as \( y = mx + c \). This format provides an easy way to understand and graph straight lines. In this equation, \( m \) represents the slope, while \( c \) is the y-intercept.

• Slope \( (m) \): This tells us how steep the line is and the direction it goes. A positive slope means the line rises as you move to the right, while a negative slope means it falls.
• Y-intercept \( (c) \): This is the point where the line crosses the y-axis. It's a crucial part of graphing because it gives a starting point for drawing the line.

Understanding this form makes it simpler to identify both the slope and the y-intercept directly from the equation. This spatial understanding allows for faster graphing without the need for additional calculations.

Y-Intercept

The y-intercept of a linear equation is particularly useful because it represents the specific point where the line crosses the y-axis. In the equation \( y = \frac{1}{3}x + 2 \), you can identify the y-intercept by looking at the constant term, \( c \), which is 2 in this case.

• Identifying the y-intercept: To find the y-intercept, check the value of \( c \) in the equation written in slope-intercept form \( y = mx + c \). Here, \( c = 2 \) indicates that the graph of the line will cross the y-axis at (0, 2).
• Significance: The y-intercept is the output value when \( x \) equals zero. It offers a quick way to start drawing the graph, even without knowing the slope.

By pinpointing the y-intercept, you have a concrete anchor for your graph, simplifying the process of plotting the rest of the line.

Graphing Linear Functions

Graphing a linear function can be made straightforward if you follow specific steps, utilizing the properties of the equation in the slope-intercept form. Here's how you can go about it:

• Start with the y-intercept: Begin by plotting the y-intercept on the graph. If your equation is \( y = \frac{1}{3}x + 2 \), plot (0, 2) on the y-axis.
• Use the slope: From the y-intercept point, use the slope \( \frac{1}{3} \) to determine the direction of the line. Since the slope is positive and corresponds to "rise over run," move up 1 unit and right 3 units to find another point on the line.
• Draw the line: After plotting at least two points, draw a straight line through them. Extend the line across the graph, with arrows on each end signifying continuation.

This method of graphing allows a clear visualization of the line represented by the equation, making it easier to analyze and understand the relationship between the variables.