Problem 4
Question
Sketch the graph of the function by first making a table of values. $$ f(x)=6-3 x $$
Step-by-Step Solution
Verified Answer
Create a table of values for \(f(x) = 6 - 3x\), plot the points, and draw a line through them.
1Step 1: Create a Table of Values
To understand the behavior of the function, we first create a table of values. Choose a set of x-values (e.g., -1, 0, 1, 2, 3) to substitute into the function and calculate their corresponding y-values using the equation \(f(x) = 6 - 3x\).
2Step 2: Substitute x-values into the Function
Calculate the y-values for each chosen x-value. For example:- For \(x = -1\), \(f(-1) = 6 - 3(-1) = 9\).- For \(x = 0\), \(f(0) = 6 - 3(0) = 6\).- For \(x = 1\), \(f(1) = 6 - 3(1) = 3\).- For \(x = 2\), \(f(2) = 6 - 3(2) = 0\).- For \(x = 3\), \(f(3) = 6 - 3(3) = -3\).
3Step 3: Compile Results into Table
Compile the x-values and their corresponding y-values into a table:
| x | f(x) |
|---|------|
| -1 | 9 |
| 0 | 6 |
| 1 | 3 |
| 2 | 0 |
| 3 | -3 |
4Step 4: Plot the Points on a Graph
Using the table of values, plot the points on a Cartesian plane. Each pair (x, f(x)) from the table becomes a point on the graph, such as (-1, 9), (0, 6), (1, 3), (2, 0), and (3, -3).
5Step 5: Draw the Line
Observe that the points form a straight line. Draw a straight line through these points to complete the graph of the linear function \(f(x) = 6 - 3x\). This line will have a slope of -3 and a y-intercept at 6.
Key Concepts
Graphing FunctionsTable of ValuesSlope and InterceptAlgebra
Graphing Functions
To graph a function means to visually represent how a particular function behaves, typically by plotting it on a coordinate grid. It shows the relationship between the independent variable (usually x) and the dependent variable (f(x) or y). For linear functions like the one in this exercise, you will always get a straight line because these functions describe constant rates of change.
Graphing begins by identifying the type of function you have. Linear functions, which take the form \(f(x) = mx + b\), are straightforward as they depict a straight line on the graph. By plotting specific points and connecting them with a line, you create a visual story of how the variables interact. In this example, we start with the linear function \(f(x) = 6 - 3x\), where each chosen pair of x and corresponding y creates a point on the graph.
By following these steps, you gain insights into the behavior of the function—such as its direction, slope, and intercepts—which aids in understanding the underlying algebraic concepts.
Graphing begins by identifying the type of function you have. Linear functions, which take the form \(f(x) = mx + b\), are straightforward as they depict a straight line on the graph. By plotting specific points and connecting them with a line, you create a visual story of how the variables interact. In this example, we start with the linear function \(f(x) = 6 - 3x\), where each chosen pair of x and corresponding y creates a point on the graph.
- Choose x-values to substitute into the function.
- Calculate corresponding y-values to form coordinate pairs.
- Plot these points on a Cartesian plane and connect them with a line.
By following these steps, you gain insights into the behavior of the function—such as its direction, slope, and intercepts—which aids in understanding the underlying algebraic concepts.
Table of Values
Creating a table of values is a systematic approach to understanding a function's behavior before graphing. This step involves selecting suitable x-values, calculating their corresponding y-values, and compiling the results in a table format. This method helps to easily find multiple points needed for graphing.
For the function \(f(x) = 6 - 3x\), a table of values might look something like this:
This table makes graphing easier by presenting selected coordinates clearly and efficiently. Each (x, f(x)) pair corresponds to a point on the graph. Students should note the importance of choosing a reasonable range of x-values to visualize different parts of the graph. By doing this, you clarify how changes in x affect the outcome of f(x).
For the function \(f(x) = 6 - 3x\), a table of values might look something like this:
- For \(x = -1\), \(f(-1) = 9\).
- For \(x = 0\), \(f(0) = 6\).
- For \(x = 1\), \(f(1) = 3\).
- For \(x = 2\), \(f(2) = 0\).
- For \(x = 3\), \(f(3) = -3\).
This table makes graphing easier by presenting selected coordinates clearly and efficiently. Each (x, f(x)) pair corresponds to a point on the graph. Students should note the importance of choosing a reasonable range of x-values to visualize different parts of the graph. By doing this, you clarify how changes in x affect the outcome of f(x).
Slope and Intercept
Understanding the slope and intercept is crucial when working with linear functions like \(f(x) = 6 - 3x\). The slope, denoted as 'm' in the formula \(f(x) = mx + b\), measures the steepness or tilt of the line, capturing how much the function's output (y) changes with a unit increase in input (x). The y-intercept, given by 'b', is where the line crosses the y-axis when \(x = 0\).
In our example function \(f(x) = 6 - 3x\), the slope is \(-3\). This indicates that for every 1 unit increase in x, the y-value decreases by 3 units. A negative slope results in a line that slants downwards from left to right. The line crosses the y-axis at 6, which is the y-intercept. This means that when x is zero, f(x) equals 6.
In our example function \(f(x) = 6 - 3x\), the slope is \(-3\). This indicates that for every 1 unit increase in x, the y-value decreases by 3 units. A negative slope results in a line that slants downwards from left to right. The line crosses the y-axis at 6, which is the y-intercept. This means that when x is zero, f(x) equals 6.
- Slope (-3): Change in y per unit change in x.
- Y-intercept (6): Point at which the line meets the y-axis.
Algebra
Algebra provides the tools needed to analyze and comprehend functions like \(f(x) = 6 - 3x\). It's all about manipulating equations and expressions to find unknowns or predict outcomes. With linear functions, algebra helps you understand how changes in x influence f(x) using basic arithmetic and algebraic principles.
By substituting various x-values, you can employ algebra to compute corresponding f(x) values, revealing the constant rate of change in linear functions. Algebra also aids in identifying crucial features of the function such as slope and y-intercept, forming the basis for further graphical insights.
The use of algebra enables students to scrutinize and interpret the mathematical relationships within functions, building a foundation for mastering advanced mathematical concepts.
By substituting various x-values, you can employ algebra to compute corresponding f(x) values, revealing the constant rate of change in linear functions. Algebra also aids in identifying crucial features of the function such as slope and y-intercept, forming the basis for further graphical insights.
- Equation Form: Transformation of functions into \(f(x) = mx + b\) format.
- Solving for y: Compute y-values for given x-values.
- Predictive Analysis: Use algebra to infer effects of variable changes on the equation outcome.
The use of algebra enables students to scrutinize and interpret the mathematical relationships within functions, building a foundation for mastering advanced mathematical concepts.
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