Using These Points, What Is The Graph Of This Function Rule?$\[ Y = 4 - 3x \\]$\[ \begin{array}{c|c} x & Y \\ \hline 0 & 4 \\ 1 & 1 \\ 2 & -2 \\ \end{array} \\]
Introduction
In mathematics, a graph is a visual representation of a function, showing the relationship between the input (x-values) and the output (y-values). When given a function rule, such as y = 4 - 3x, we can use various methods to determine its graph. In this article, we will explore how to graph a linear function using a table of values and discuss the key characteristics of the resulting graph.
The Function Rule
The given function rule is y = 4 - 3x. This is a linear function, which means it has a constant rate of change between any two points on the graph. The general form of a linear function is y = mx + b, where m is the slope and b is the y-intercept.
The Table of Values
To create a table of values, we substitute different x-values into the function rule and calculate the corresponding y-values. The table of values is shown below:
| x | y |
|---|---|
| 0 | 4 |
| 1 | 1 |
| 2 | -2 |
Analyzing the Table of Values
From the table of values, we can see that as x increases by 1, y decreases by 3. This indicates a negative slope, which means the graph will slope downward from left to right.
The Graph of the Function
Using the table of values, we can plot the points (0, 4), (1, 1), and (2, -2) on a coordinate plane. By connecting these points with a straight line, we can visualize the graph of the function.
Key Characteristics of the Graph
The graph of the function y = 4 - 3x has several key characteristics:
- Slope: The slope of the graph is -3, which means it slopes downward from left to right.
- Y-intercept: The y-intercept is 4, which means the graph crosses the y-axis at the point (0, 4).
- X-intercept: To find the x-intercept, we set y = 0 and solve for x. This gives us the equation 0 = 4 - 3x, which simplifies to 3x = 4 and x = 4/3. Therefore, the x-intercept is (4/3, 0).
- Domain and Range: The domain of the function is all real numbers, and the range is all real numbers less than or equal to 4.
Conclusion
In conclusion, the graph of the function y = 4 - 3x is a straight line with a negative slope and a y-intercept of 4. The graph slopes downward from left to right and has an x-intercept at (4/3, 0). By analyzing the table of values and plotting the points on a coordinate plane, we can visualize the graph of the function and understand its key characteristics.
Real-World Applications
Linear functions have many real-world applications, including:
- Cost and Revenue Analysis: Linear functions can be used to model the cost and revenue of a business, helping to determine the break-even point and optimal pricing strategy.
- Distance and Time: Linear functions can be used to model the distance traveled by an object over time, helping to determine the speed and direction of the object.
- Finance: Linear functions can be used to model the growth of an investment over time, helping to determine the future value of the investment.
Tips for Graphing Linear Functions
When graphing linear functions, keep the following tips in mind:
- Use a table of values: A table of values can help you visualize the graph of the function and identify key characteristics such as the slope and y-intercept.
- Plot points on a coordinate plane: Plotting points on a coordinate plane can help you visualize the graph of the function and understand its key characteristics.
- Use a straight line: Linear functions have a constant rate of change, so the graph will be a straight line.
- Identify the slope and y-intercept: The slope and y-intercept are key characteristics of the graph, and can be used to determine the equation of the line.
Common Mistakes to Avoid
When graphing linear functions, avoid the following common mistakes:
- Incorrect slope: Make sure to calculate the slope correctly, using the formula m = (y2 - y1) / (x2 - x1).
- Incorrect y-intercept: Make sure to calculate the y-intercept correctly, using the formula b = y1 - m(x1).
- Incorrect graph: Make sure to plot the points correctly and draw a straight line through them.
Conclusion
Q: What is a linear function?
A: A linear function is a function that has a constant rate of change between any two points on the graph. It can be represented by the equation y = mx + b, where m is the slope and b is the y-intercept.
Q: What is the slope of a linear function?
A: The slope of a linear function is the rate of change between any two points on the graph. It can be calculated using the formula m = (y2 - y1) / (x2 - x1).
Q: What is the y-intercept of a linear function?
A: The y-intercept of a linear function is the point where the graph crosses the y-axis. It can be calculated using the formula b = y1 - m(x1).
Q: How do I graph a linear function?
A: To graph a linear function, follow these steps:
- Create a table of values by substituting different x-values into the function rule and calculating the corresponding y-values.
- Plot the points on a coordinate plane.
- Draw a straight line through the points.
Q: What are some common mistakes to avoid when graphing linear functions?
A: Some common mistakes to avoid when graphing linear functions include:
- Incorrect slope
- Incorrect y-intercept
- Incorrect graph
Q: How do I determine the equation of a linear function?
A: To determine the equation of a linear function, follow these steps:
- Identify the slope (m) and y-intercept (b) of the graph.
- Use the equation y = mx + b to write the equation of the line.
Q: What are some real-world applications of linear functions?
A: Some real-world applications of linear functions include:
- Cost and revenue analysis
- Distance and time
- Finance
Q: How do I use a table of values to graph a linear function?
A: To use a table of values to graph a linear function, follow these steps:
- Create a table of values by substituting different x-values into the function rule and calculating the corresponding y-values.
- Plot the points on a coordinate plane.
- Draw a straight line through the points.
Q: What is the difference between a linear function and a nonlinear function?
A: A linear function has a constant rate of change between any two points on the graph, while a nonlinear function does not have a constant rate of change.
Q: How do I determine if a function is linear or nonlinear?
A: To determine if a function is linear or nonlinear, follow these steps:
- Create a table of values by substituting different x-values into the function rule and calculating the corresponding y-values.
- Plot the points on a coordinate plane.
- If the graph is a straight line, the function is linear. If the graph is not a straight line, the function is nonlinear.
Q: What are some common types of linear functions?
A: Some common types of linear functions include:
- Slope-intercept form (y = mx + b)
- Point-slope form (y - y1 = m(x - x1))
- Standard form (Ax + By = C)
Q: How do I convert between different forms of linear functions?
A: To convert between different forms of linear functions, follow these steps:
- Identify the form of the function you want to convert to.
- Use the appropriate formula to convert the function.
Q: What are some common applications of linear functions in real-world scenarios?
A: Some common applications of linear functions in real-world scenarios include:
- Cost and revenue analysis
- Distance and time
- Finance
- Science and engineering
Conclusion
In conclusion, graphing linear functions is an important skill in mathematics, with many real-world applications. By understanding the key characteristics of linear functions, such as the slope and y-intercept, and using a table of values to graph the function, you can visualize the graph of a linear function and understand its key characteristics. Remember to avoid common mistakes such as incorrect slope and y-intercept, and to use a straight line to represent the graph.