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Chapter 2: Problem 16

Sketch the graph of each equation. $$ k(x)=\frac{2}{3} x-3 $$

Short Answer

Expert verified
The graph of \( k(x) = \frac{2}{3}x - 3 \) is a straight line with y-intercept \((0, -3)\) and passes through \((3, -1)\).

Step by step solution

01

Identify the Type of Equation

The equation given is \( k(x) = \frac{2}{3}x - 3 \). This is a linear equation of the form \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept. In this case, \( m = \frac{2}{3} \) and \( c = -3 \).
02

Determine the Y-intercept

The y-intercept is the point where the graph crosses the y-axis. This occurs when \( x = 0 \). Substituting \( x = 0 \) into the equation \( k(x) = \frac{2}{3}x - 3 \) gives \( k(0) = -3 \). Therefore, the y-intercept is \( (0, -3) \).
03

Determine the Slope

The slope \( m = \frac{2}{3} \) indicates that for every increase of 3 units in \( x \), \( y \) increases by 2 units. This means the rise over run is \( \frac{2}{3} \).
04

Find Another Point Using the Slope

Starting from the y-intercept \((0, -3)\), use the slope to find another point on the line. From \( (0, -3) \), move 3 units to the right (positive direction along the x-axis) and 2 units up (positive direction along the y-axis) to reach the point \( (3, -1) \).
05

Sketch the Graph

Plot the points \((0, -3)\) and \((3, -1)\) on a graph. Draw a line through these points, extending it in both directions to cover all values of \( x \). This line represents the graph of the equation \( k(x) = \frac{2}{3}x - 3 \).

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Key Concepts

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

Slope-Intercept Form
Slope-intercept form is a fundamental concept for graphing linear equations. It is expressed as \( y = mx + c \), where \( m \) represents the slope of the line, and \( c \) is the y-intercept. This form is particularly useful because it provides two key pieces of information at a glance:
  • The slope \( m \), which indicates the steepness and direction of the line.
  • The y-intercept \( c \), which tells us where the line crosses the vertical y-axis.
In the equation we are analyzing, \( k(x) = \frac{2}{3}x - 3 \), it clearly fits the slope-intercept form, making it straightforward to graph.
Y-Intercept
The y-intercept of a linear equation is crucial when graphing because it provides a starting point on the graph. It is where the line crosses the y-axis, which is the vertical axis. Finding the y-intercept involves setting the value of \( x \) to zero and solving for \( y \).

For the equation \( k(x) = \frac{2}{3}x - 3 \), substituting \( x = 0 \) yields \( k(0) = -3 \). Thus, the y-intercept is \( (0, -3) \). This point is not only easy to identify but also essential for beginning the process of graphing the line.
Graphing Techniques
When it comes to graphing linear equations, a systematic approach can simplify the process. Here's a handy technique to ensure a clear and precise graph:
  • Start by identifying the y-intercept. For the equation \( k(x) = \frac{2}{3}x - 3 \), plot the point \( (0, -3) \) on the y-axis.
  • Use the slope to determine another point. From the y-intercept, move according to the slope; in this case, \( \frac{2}{3} \) means up 2 units and 3 units to the right.
  • Plot this new point, \( (3, -1) \), on the graph.
  • Draw a straight line through the plotted points, extending it in both directions to cover a broad range of \( x \) values.
These elements combine to create a consistent method of graphing that can be applied to any linear equation in slope-intercept form.
Slope Calculation
Understanding how to calculate the slope is key to mastering linear equations. The slope is the measure of how steep a line is and its direction, calculated as the 'rise over run'.
  • In our example, \( m = \frac{2}{3} \), meaning for every 3 units you move to the right along the x-axis (run), the line goes up 2 units (rise) along the y-axis.
  • The slope is positive, indicating the line ascends as you move from left to right.
Grasping this concept means you can quickly discern the nature of a line and predict its behavior across a graph.

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