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Chapter 4: Problem 93

Find a linear function whose graph has slope \(\frac{1}{3}\) and \(y\) -intercept \((0,-2)\)

Short Answer

Expert verified
The linear function is \( y = \frac{1}{3}x - 2 \).

Step by step solution

01

Understand the Slope-Intercept Form

The slope-intercept form of a linear equation is given by:\[ y = mx + b \]where \( m \) is the slope and \( b \) is the y-intercept.
02

Identify Given Values

From the problem, the given slope \( m \) is \( \frac{1}{3} \) and the y-intercept \( b \) is \( -2 \).
03

Substitute the Values

Substitute the slope \( m = \frac{1}{3} \) and the y-intercept \( b = -2 \) into the slope-intercept form:\[ y = \frac{1}{3}x - 2 \]
04

Write the Linear Function

The linear function that satisfies the given conditions is:\[ y = \frac{1}{3}x - 2 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form
The slope-intercept form is a way of writing linear equations. It makes it easier to understand how the line behaves. This form is written as \[ y = mx + b \] This equation has two main components:
  • Slope (m): This shows the steepness of the line. It's how much the line goes up or down.

  • Y-intercept (b): This is where the line crosses the y-axis. It gives a starting point for the line.
When you know the slope and the y-intercept, you can easily draw the line or understand how it changes.
slope
The slope is a crucial part of a linear function.
It indicates how steep or flat the line is. Mathematically, the slope (\(m\)) is the ratio of the rise (change in y) over the run (change in x). For example, a slope of 1/3 means:
  • For every 3 units you move horizontally (run), the line moves up by 1 unit (rise).
Think of slope as the 'tilt' of the line:
  • If the slope is positive, the line goes upwards as you move from left to right.

  • If the slope is negative, the line goes downwards as you go from left to right.

  • A slope of zero means the line is flat (horizontal).

  • An undefined slope (like when x is constant) means the line is vertical.
y-intercept
The y-intercept (\(b\)) is another key component of a linear function. This tells you the point where the line crosses the y-axis.
For instance, if a linear function has a y-intercept of -2, it means the line crosses the y-axis at (0, -2).

The y-intercept provides a starting point:
  • It allows you to visualize where the line begins on the graph.
  • Knowing the y-intercept, you can start drawing the line directly from this point.
Combined with the slope, the y-intercept helps define the entire line. As in the given exercise, the slope is 1/3, and the y-intercept is -2. So the line equation is: \[ y = \frac{1}{3}x - 2 \]

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