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Chapter 3: Problem 7

Sketch the graph of the function by first making a table of values. \(f(x)=2 x-4\)

Short Answer

Expert verified
Plot points for \(x = -2, -1, 0, 1, 2\), then draw a line through them.

Step by step solution

01

Choose Values for x

To begin sketching the graph of the function, select a range of values for \(x\). This range should cover both negative and positive numbers to provide a comprehensive view of the graph. We will choose \(-2, -1, 0, 1, 2\) as our values for \(x\).
02

Calculate Corresponding f(x) Values

Substitute each chosen \(x\) value into the function \(f(x) = 2x - 4\) to find the corresponding \(f(x)\) values. 1. For \(x = -2\): \(f(-2) = 2(-2) - 4 = -4 - 4 = -8\) 2. For \(x = -1\): \(f(-1) = 2(-1) - 4 = -2 - 4 = -6\) 3. For \(x = 0\): \(f(0) = 2(0) - 4 = 0 - 4 = -4\) 4. For \(x = 1\): \(f(1) = 2(1) - 4 = 2 - 4 = -2\) 5. For \(x = 2\): \(f(2) = 2(2) - 4 = 4 - 4 = 0\)
03

Create a Table of Values

Now, organize the \(x\) values and their corresponding \(f(x)\) values into a table:\(\begin{array}{c|c} x & f(x) \\hline-2 & -8 \-1 & -6 \ 0 & -4 \ 1 & -2 \ 2 & 0 \\end{array}\)This table will help us plot points on a coordinate plane.
04

Plot Points on a Coordinate Plane

Use the values from the table to plot points on a coordinate plane as follows: - Plot the point (-2, -8). - Plot the point (-1, -6). - Plot the point (0, -4). - Plot the point (1, -2). - Plot the point (2, 0). These points represent the graph of the function.
05

Draw the Line

Once the points are plotted, draw a straight line through them. This line represents the function \(f(x) = 2x - 4\). Ensure the line extends in both directions beyond the plotted points to indicate it continues indefinitely.

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Key Concepts

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

Making a Table of Values
In order to graph a linear function, creating a table of values is a particularly useful step. This involves selecting a set of values for the variable \(x\), then calculating the corresponding \(f(x)\) values using the given function. In our example with the function \(f(x) = 2x - 4\), we chose the \(x\)-values \(-2, -1, 0, 1, 2\).
  • Start by selecting a mix of positive and negative numbers as well as zero to get a comprehensive idea of how the graph behaves.
  • Substitute each \(x\) into the linear equation to find its corresponding \(f(x)\).
By organizing these \(x\) and \(f(x)\) pairs into a table, you create a visual aid that simplifies the plotting process. A clear table helps set the stage for visualizing how the equation transforms into a linear graph.
Plotting Points on a Graph
Once you have your table of values, the next step is to plot these points on a graph. Placing them on a grid helps to visualize the relationship defined by the linear function. On the coordinate plane:
  • Each point is an \((x, f(x))\) or \((x, y)\) pair, as derived from our table.
  • Plot these points on the graph, marking where each pair lands on the grid.
For instance, in our solution:
  • Plot point \((-2, -8)\) by moving left 2 units and down 8 units from the origin.
  • Do this for all other values: \((-1, -6)\), \((0, -4)\), \((1, -2)\), and \((2, 0)\).
Accurate plotting ensures that the line you draw through these points truly represents the function.
Understanding the Coordinate Plane
The coordinate plane, sometimes called the Cartesian plane, is a two-dimensional plane featuring a horizontal axis (x-axis) and a vertical axis (y-axis), which intersect at the origin. It is crucial for graphing linear functions:
  • The x-axis represents the independent variable, \(x\).
  • The y-axis represents the dependent variable, \(f(x)\) or \(y\).
Each point on this plane is specified by an \((x, y)\) pair, allowing us to visualize relationships between variables. In the context of linear functions, plotting the points calculated from our table of values helps reveal the nature of the equation. Understanding the coordinate plane's structure is essential for graphing and interpreting linear functions.
Exploring Linear Equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations graph as straight lines and are defined by expressions such as \(f(x) = 2x - 4\). Let's break it down:
  • The \(x\)-coefficient, 2, indicates the slope of the line, or how steep it is.
  • The constant term, -4, represents the y-intercept, i.e., where the line crosses the y-axis.
Linear equations exhibit a constant rate of change, which is why the graph is a straight line. By changing the values of \(x\) using the function, we can illustrate how the line behaves across different regions of the coordinate plane. Understanding the characteristics of linear equations simplifies the graphing process and interpretation of data in real-world contexts.

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Most popular questions from this chapter

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