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Chapter 12: Problem 23

Graph each function. $$g(x)=-\frac{3}{2} x-1$$

Short Answer

Expert verified
To graph the function \(g(x) = -\frac{3}{2}x -1\), first identify the slope (m) as -\(\frac{3}{2}\) and the y-intercept (b) as -1. Plot the y-intercept point (0, -1) on the graph. Use the slope to find another point on the line by going down 3 units and right 2 units to get the point (2, -4). Draw a straight line through both points, representing the graph of the given function.

Step by step solution

01

Identify the slope and y-intercept

The given function is in slope-intercept form (\(y = mx + b\)), where m is the slope and b is the y-intercept. From the function \(g(x) = -\frac{3}{2}x -1\), we can identify the slope (m) and y-intercept (b) as follows: Slope (m) = -\(\frac{3}{2}\) Y-intercept (b) = -1
02

Plot the y-intercept on the graph

The y-intercept is the point where the line crosses the y-axis. In this case, the y-intercept is -1, so the point on the graph will be (0, -1). Plot this point on the graph.
03

Use the slope to find another point on the line

To use the slope to find another point on the line, keep in mind that the slope is the change in y divided by the change in x, or \(\frac{rise}{run}\). In this case, the slope is -\(\frac{3}{2}\), which means we can go down 3 units in the y-axis and to the right 2 units in the x-axis, starting from the y-intercept. From the y-intercept point (0, -1), move 3 units down and 2 units to the right. The new point will be (2, -4). Plot this point on the graph.
04

Draw the line through the points

Now that we have the y-intercept and another point on the line, connect the two points with a straight line. This line represents the graph of the function \(g(x) = -\frac{3}{2}x -1\).

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Key Concepts

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

Graphing Techniques
Graphing linear functions on a coordinate plane involves using specific techniques to accurately represent the data. The most common approach is to utilize points that can be derived from the slope-intercept form of a function. There are a few essential steps to follow:
  • Identify the key points of the linear function, starting with the y-intercept where the line will initially cross the y-axis.
  • Use the slope to locate additional points that lie on the line by moving from one point to another according to the rise and run.
  • Connect these points using a straight edge for an accurate line representation.
These techniques ensure a correct depiction of a linear function, which is essential for analysis and interpretation when working with linear models.
Slope-Intercept Form
The slope-intercept form of a linear equation is written as \(y = mx + b\). This form is particularly useful for graphing and understanding linear relationships. Here's what each part stands for:
  • \(m\) represents the slope of the line, indicating its steepness and direction.
  • \(b\) is the y-intercept, the point where the line meets the y-axis.
Using the equation \(g(x) = -\frac{3}{2}x - 1\), we determine that the slope, \(m\), is \(-\frac{3}{2}\) and the y-intercept, \(b\), is \(-1\). This means that as x increases by 2 units, y decreases by 3 units, giving a downward slope. The y-intercept tells us the line crosses the y-axis at -1. This form simplifies the process of graphing as it directly gives the slope and starting point of the graph.
Coordinate Plane
The coordinate plane is a two-dimensional area formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). These axes intersect at the origin, denoted as (0,0), splitting the plane into four quadrants. Here's a quick breakdown:
  • The first quadrant is where both x and y are positive.
  • The second quadrant has negative x and positive y values.
  • The third quadrant includes negative x and y values.
  • The fourth quadrant features positive x and negative y values.
Graphing on a coordinate plane involves plotting points given their x (horizontal) and y (vertical) coordinates, such as (0,-1) and (2,-4) from the function \(g(x)=-\frac{3}{2}x-1\). This plane is fundamental in visualizing mathematical concepts, providing a clear view of how linear functions behave.
Plotting Points
Plotting points on a coordinate plane is a core skill in graphing linear functions. Each point is defined by a pair of numbers known as coordinates, written as (x, y). Here's a simple procedure for plotting points:
  • Start at the origin, (0,0), where the axes intersect.
  • For the x-coordinate, move right (positive) or left (negative) along the x-axis.
  • For the y-coordinate, move up (positive) or down (negative) along the y-axis.
For example, to plot the point (0,-1), you remain at the origin for x, but move down 1 unit on the y-axis. For (2,-4), you move 2 units to the right and then 4 units down. Once points are plotted, lines can be drawn through them, representing the linear equation \(y = -\frac{3}{2}x - 1\). This process illustrates the relationship described by the function through visual means.

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Most popular questions from this chapter

Graph the following piecewise functions. $$k(x)=\left\\{\begin{array}{cc}x+1, & x \geq-2 \\\2 x+8, & x<-2\end{array}\right.$$

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