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Magazine Chapter 1: Problem 21
Sketch the graph of \(f\) by hand. Do not use a calculator. $$f(x)=\frac{1}{2} x$$
Short Answer
Expert verified
The graph is a straight line passing through the origin with a slope of \(\frac{1}{2}\).
Step by step solution
01
Identify the Function Type
The function given is a linear function of the form \( f(x) = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Here, \( m = \frac{1}{2} \) and \( b = 0 \). This means the graph of the function will be a straight line.
02
Determine the Slope and Y-intercept
The slope of the function \( f(x) = \frac{1}{2}x \) is \( \frac{1}{2} \), which means for every 1 unit increase in \( x \), \( y \) increases by \( \frac{1}{2} \). The y-intercept is 0, which means the line passes through the origin (0, 0).
03
Plot the Y-intercept
Plot the point (0, 0) on a coordinate plane. This point is where the line will intersect the y-axis.
04
Use the Slope to Find Another Point
Starting from the y-intercept (0, 0), use the slope to find another point on the line. Since the slope is \( \frac{1}{2} \), move 1 unit to the right and \( \frac{1}{2} \) unit up to locate the next point, which is (1, \( \frac{1}{2} \)). Plot this point.
05
Draw the Line
Draw a straight line through the points (0, 0) and (1, \( \frac{1}{2} \)). This line represents the graph of the function \( f(x) = \frac{1}{2}x \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Graphing Techniques
Graphing a linear function by hand can be made simple by following a few straightforward steps. First, start by identifying the type of function you have. In this case, we have a linear function. A linear function, as the name implies, will graph as a straight line on the coordinate plane. The next step is to gather information about the slope and y-intercept, which will guide how you sketch your graph.
### Steps for Graphing
- **Identify the points**: These help you position your line accurately. The key points you will need are the y-intercept and a point derived from the slope.
- **Draw using a ruler**: Straight lines should indeed be straight. Ensure that you connect your points properly.
Slope-Intercept Form
The slope-intercept form is a popular form of expressing linear functions. It comes in the format of \( y = mx + b \), where **\( m \)** is the slope and **\( b \)** is the y-intercept. This form is invaluable because it immediately tells you two critical pieces of information: how steep the line is (the slope), and where it crosses the y-axis (the y-intercept). ### Slope (\( m \))
- **Definition**: The slope represents how much the line "rises" vertically for each unit it moves "across" horizontally.
- **Example**: For the function \( f(x) = \frac{1}{2}x \), the slope is \( \frac{1}{2} \). This means that for a movement of 1 unit horizontally, the line moves up by \( \frac{1}{2} \) units.
- **Definition**: The point where the line crosses the y-axis.
- **Example**: In \( f(x) = \frac{1}{2}x \), the y-intercept is 0, implying the line passes through the origin (0, 0).
Coordinate Plane
The coordinate plane is a two-dimensional surface on which we can graphically represent equations. It is divided into four quadrants by the x-axis (horizontal) and the y-axis (vertical). Each point on the plane is described by a pair of numbers \((x, y)\), known as coordinates. ### Features of the Coordinate Plane
- **Axes**: The x-axis runs horizontally, and the y-axis runs vertically. Their intersection point is known as the origin (0,0).
- **Quadrants**: The plane is divided into four quadrants: I, II, III, and IV, moving counterclockwise from the upper right.
- **Plotting Points**: Each point represented as \((x, y)\) can be accurately located on the plane by moving \(x\) units along the x-axis and \(y\) units along the y-axis.
