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Chapter 4: Problem 28

Graph each function by making a table of values and plotting points. $$g(x)=-\frac{3}{5} x+2$$

Short Answer

Expert verified
To graph \(g(x) = -\frac{3}{5}x + 2\), we first create a table of values using \(x = -5, -2, 0, 3, 5\), which results in the points (-5, 7), (-2, 4.4), (0, 2), (3, 0.2), and (5, -1). We then plot these points on a coordinate plane and connect them with a straight line to form the graph of the function.

Step by step solution

01

Create a table of values

Choose several values of \(x\) to plug into the function equation \(g(x) = -\frac{3}{5}x + 2\). In general, choose values that include both positive and negative numbers. Here, we will use \(x = -5, -2, 0, 3, 5\). Plug in these values one at a time and solve for \(g(x)\) (the \(y\) values).
02

Determine the corresponding y values

For each chosen value of \(x\), plug it into the function equation and solve for the corresponding \(y\) value: 1. For \(x = -5\), \(g(-5) = -\frac{3}{5}(-5) + 2 = 5+2=7\) 2. For \(x = -2\), \(g(-2) = -\frac{3}{5}(-2) + 2 = 2.4+2=4.4\) 3. For \(x = 0\), \(g(0) = -\frac{3}{5}(0) + 2 = 0+2=2\) 4. For \(x = 3\), \(g(3) = -\frac{3}{5}(3) + 2 = -1.8+2=0.2\) 5. For \(x = 5\), \(g(5) = -\frac{3}{5}(5) + 2 = -3+2=-1\) Now we have our table of values: | x | g(x) | |-----|-------| | -5 | 7.0 | | -2 | 4.4 | | 0 | 2.0 | | 3 | 0.2 | | 5 | -1.0 |
03

Plot the points on a coordinate plane

Using the table of values, plot each point on a coordinate plane. The points will be as follows: - (-5, 7) - (-2, 4.4) - (0, 2) - (3, 0.2) - (5, -1)
04

Connect the plotted points

Connect the plotted points with a straight line, as this is a linear function. The line should pass through all the plotted points, clearly showing the graph of the given function \(g(x) = -\frac{3}{5}x + 2\). And now, the graph of the function is complete.

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Key Concepts

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

Linear Functions
A linear function is a type of function where each output value is directly related to its input value by a constant rate of change. This means that when you graph a linear function, you'll get a straight line. The general form of a linear function is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
  • The slope \(m\) indicates how steep the line is. It's the ratio of the change in the y-values to the change in the x-values.
  • The y-intercept \(b\) is the point where the line crosses the y-axis.
For example, in our function \(g(x) = -\frac{3}{5}x + 2\), the slope is \(-\frac{3}{5}\) and the y-intercept is 2. This tells us the line will slant downwards and cross the y-axis at the point \((0, 2)\).
Table of Values
When graphing functions, like our linear function, creating a table of values is a valuable first step. This table helps organize specific input-output pairs, making it easier to plot the graph.
  • Choose several values for \(x\). Some should be negative, some positive, and include zero.
  • Plug each \(x\) value into the function to find the corresponding \(g(x)\) or \(y\) value.
In our example, we selected the values \(-5, -2, 0, 3, 5\) for \(x\), calculated the \(g(x)\) for each, and got the values \(7, 4.4, 2, 0.2, -1\), respectively. This collection of pairs reveals the behavior of the linear function and serves as the basis for plotting it on the graph.
Coordinate Plane
The coordinate plane is a two-dimensional surface where you can plot points, lines, and curves to visually represent functions. It consists of two axes:
  • The x-axis is horizontal and represents input values.
  • The y-axis is vertical and represents output values.
The point where these two axes meet is called the origin, denoted by \((0, 0)\). In graphing the function \(g(x) = -\frac{3}{5}x + 2\), points from the table of values are plotted on the coordinate plane. This plane provides a clear visual depiction of how the value of \(g(x)\) changes with \(x\).
Plotting Points
Plotting points on the coordinate plane is the next crucial step in graphing a function. Each point corresponds to an input-output pair from the table of values.
  • Locate the x-value on the x-axis.
  • Find the corresponding y-value on the y-axis.
  • Mark the intersection of the two as the point \((x, y)\).
In our function, points like \((-5, 7)\), \((-2, 4.4)\), \((0, 2)\), \((3, 0.2)\), and \((5, -1)\) are plotted. Once all points are plotted, draw a straight line through them to represent the graph of the function. This visual approach simplifies understanding the relationship between the variables in the function.

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Most popular questions from this chapter

Lines \(L_{1}\) and \(L_{2}\) contain the given points. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. $$\begin{aligned}&L_{1}:(-3,9),(4,2)\\\&L_{2}:(6,-8),(-10,8)\end{aligned}$$

Lines \(L_{1}\) and \(L_{2}\) contain the given points. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. $$\begin{aligned}&L_{1}:(5,-1),(7,3)\\\&L_{2}:(-6,0),(4,5)\end{aligned}$$

Determine the domain of each relation, and determine whether each relation describes \(y\) as a function of \(x .\) $$y=\frac{1}{-6+4 x}$$

Write the slope-intercept form (if possible) of the equation of the line meeting the given conditions. perpendicular to \(y=\frac{3}{4}\) containing \((-2,5)\)

Lines \(L_{1}\) and \(L_{2}\) contain the given points. Determine if lines \(L_{1}\) and \(L_{2}\) are parallel, perpendicular, or neither. $$\begin{aligned}&L_{1}:(-3,2),(0,2)\\\&L_{2}:(1,-1),(-2,-1)\end{aligned}$$
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