Misty Correctly Determined The Equation Of The Linear Function Represented By The Table Of Values Below To Be $y = -2x + 9$ In Slope-intercept Form By Using The Ordered Pairs $(1,7)$ And
Introduction
Linear functions are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, economics, and computer science. A linear function is a polynomial function of degree one, which means it has a single variable raised to the power of one. In this article, we will explore the concept of linear functions, their representation in slope-intercept form, and how to determine the equation of a linear function using a table of values.
What is a Linear Function?
A linear function is a function that can be written in the form:
y = mx + b
where m is the slope of the line, and b is the y-intercept. The slope of a line is a measure of how steep it is, and it is calculated as the ratio of the vertical change (rise) to the horizontal change (run). The y-intercept is the point where the line intersects the y-axis.
Slope-Intercept Form
The slope-intercept form of a linear function is a way of writing the equation of a line in terms of its slope and y-intercept. It is the most common form of a linear function, and it is used to represent lines in a variety of contexts.
Determining the Equation of a Linear Function
To determine the equation of a linear function, we need to use a table of values that represents the function. A table of values is a list of ordered pairs that show the input values (x) and the corresponding output values (y).
Example: Determining the Equation of a Linear Function
Let's consider the table of values below:
| x | y |
|---|---|
| 1 | 7 |
| 2 | 5 |
| 3 | 3 |
| 4 | 1 |
To determine the equation of the linear function represented by this table of values, we need to find the slope (m) and the y-intercept (b).
Finding the Slope
To find the slope, we need to calculate the ratio of the vertical change (rise) to the horizontal change (run). We can do this by selecting two points from the table of values and calculating the slope using the formula:
m = (y2 - y1) / (x2 - x1)
Let's select the points (1, 7) and (2, 5). Plugging these values into the formula, we get:
m = (5 - 7) / (2 - 1) m = -2 / 1 m = -2
Finding the Y-Intercept
To find the y-intercept, we need to find the point where the line intersects the y-axis. We can do this by substituting x = 0 into the equation of the line. Since we know that the slope is -2, we can write the equation of the line as:
y = -2x + b
Substituting x = 0, we get:
y = -2(0) + b y = b
Since the point (0, b) is the y-intercept, we can find the value of b by substituting x = 0 into the table of values. From the table, we can see that the point (0, 9) is not present, but we can use the point (1, 7) to find the value of b.
Substituting x = 1 and y = 7 into the equation of the line, we get:
7 = -2(1) + b 7 = -2 + b b = 9
The Equation of the Linear Function
Now that we have found the slope (m = -2) and the y-intercept (b = 9), we can write the equation of the linear function in slope-intercept form:
y = -2x + 9
This is the equation of the linear function represented by the table of values.
Conclusion
In this article, we have explored the concept of linear functions, their representation in slope-intercept form, and how to determine the equation of a linear function using a table of values. We have seen how to find the slope and the y-intercept of a linear function, and how to write the equation of the function in slope-intercept form. With this knowledge, you can now determine the equation of a linear function using a table of values.
Frequently Asked Questions
Q: What is a linear function?
A: A linear function is a polynomial function of degree one, which means it has a single variable raised to the power of one.
Q: What is the slope-intercept form of a linear function?
A: The slope-intercept form of a linear function is a way of writing the equation of a line in terms of its slope and y-intercept.
Q: How do I determine the equation of a linear function using a table of values?
A: To determine the equation of a linear function using a table of values, you need to find the slope and the y-intercept of the function.
Q: How do I find the slope of a linear function?
A: To find the slope of a linear function, you need to calculate the ratio of the vertical change (rise) to the horizontal change (run).
Q: How do I find the y-intercept of a linear function?
A: To find the y-intercept of a linear function, you need to find the point where the line intersects the y-axis.
Q: What is the equation of the linear function represented by the table of values below?
A: The equation of the linear function represented by the table of values below is y = -2x + 9.
References
- [1] Khan Academy. (n.d.). Linear Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f6c/x2f1f7c/x2f1f7d
- [2] Math Open Reference. (n.d.). Linear Functions. Retrieved from https://www.mathopenref.com/linfunc.html
- [3] Wolfram MathWorld. (n.d.). Linear Function. Retrieved from https://mathworld.wolfram.com/LinearFunction.html
Glossary
- Linear function: A polynomial function of degree one, which means it has a single variable raised to the power of one.
- Slope-intercept form: A way of writing the equation of a line in terms of its slope and y-intercept.
- Slope: A measure of how steep a line is, calculated as the ratio of the vertical change (rise) to the horizontal change (run).
- Y-intercept: The point where the line intersects the y-axis.
Frequently Asked Questions: Linear Functions =============================================
Q: What is a linear function?
A: A linear function is a polynomial function of degree one, which means it has a single variable raised to the power of one. It can be written in the form y = mx + b, where m is the slope and b is the y-intercept.
Q: What is the slope-intercept form of a linear function?
A: The slope-intercept form of a linear function is a way of writing the equation of a line in terms of its slope and y-intercept. It is the most common form of a linear function and is used to represent lines in a variety of contexts.
Q: How do I determine the equation of a linear function using a table of values?
A: To determine the equation of a linear function using a table of values, you need to find the slope and the y-intercept of the function. You can do this by selecting two points from the table of values and calculating the slope using the formula m = (y2 - y1) / (x2 - x1). Then, you can use the point-slope form of a linear equation to find the equation of the line.
Q: How do I find the slope of a linear function?
A: To find the slope of a linear function, you need to calculate the ratio of the vertical change (rise) to the horizontal change (run). You can do this by selecting two points from the table of values and calculating the slope using the formula m = (y2 - y1) / (x2 - x1).
Q: How do I find the y-intercept of a linear function?
A: To find the y-intercept of a linear function, you need to find the point where the line intersects the y-axis. You can do this by substituting x = 0 into the equation of the line. Since the point (0, b) is the y-intercept, you can find the value of b by substituting x = 0 into the table of values.
Q: What is the equation of the linear function represented by the table of values below?
| x | y |
|---|---|
| 1 | 7 |
| 2 | 5 |
| 3 | 3 |
| 4 | 1 |
A: The equation of the linear function represented by the table of values below is y = -2x + 9.
Q: How do I graph a linear function?
A: To graph a linear function, you need to plot the points on a coordinate plane and draw a line through them. You can also use the slope-intercept form of a linear equation to graph the line.
Q: What is the difference between a linear function and a quadratic function?
A: A linear function is a polynomial function of degree one, while a quadratic function is a polynomial function of degree two. A linear function has a single variable raised to the power of one, while a quadratic function has a single variable raised to the power of two.
Q: Can a linear function have a negative slope?
A: Yes, a linear function can have a negative slope. A negative slope indicates that the line slopes downward from left to right.
Q: Can a linear function have a zero slope?
A: Yes, a linear function can have a zero slope. A zero slope indicates that the line is horizontal.
Q: Can a linear function have a positive slope?
A: Yes, a linear function can have a positive slope. A positive slope indicates that the line slopes upward from left to right.
Q: Can a linear function have a y-intercept of zero?
A: Yes, a linear function can have a y-intercept of zero. A y-intercept of zero indicates that the line intersects the y-axis at the origin.
Q: Can a linear function have a y-intercept of a non-zero value?
A: Yes, a linear function can have a y-intercept of a non-zero value. A non-zero y-intercept indicates that the line intersects the y-axis at a point other than the origin.
Q: Can a linear function be represented by a table of values?
A: Yes, a linear function can be represented by a table of values. A table of values is a list of ordered pairs that show the input values (x) and the corresponding output values (y).
Q: Can a linear function be represented by an equation?
A: Yes, a linear function can be represented by an equation. An equation is a statement that two expressions are equal.
Q: Can a linear function be represented by a graph?
A: Yes, a linear function can be represented by a graph. A graph is a visual representation of a function.
Q: Can a linear function be represented by a table of values and an equation?
A: Yes, a linear function can be represented by a table of values and an equation. A table of values is a list of ordered pairs that show the input values (x) and the corresponding output values (y), while an equation is a statement that two expressions are equal.
Q: Can a linear function be represented by a graph and an equation?
A: Yes, a linear function can be represented by a graph and an equation. A graph is a visual representation of a function, while an equation is a statement that two expressions are equal.
Q: Can a linear function be represented by a table of values, a graph, and an equation?
A: Yes, a linear function can be represented by a table of values, a graph, and an equation. A table of values is a list of ordered pairs that show the input values (x) and the corresponding output values (y), a graph is a visual representation of a function, and an equation is a statement that two expressions are equal.
References
- [1] Khan Academy. (n.d.). Linear Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f6c/x2f1f7c/x2f1f7d
- [2] Math Open Reference. (n.d.). Linear Functions. Retrieved from https://www.mathopenref.com/linfunc.html
- [3] Wolfram MathWorld. (n.d.). Linear Function. Retrieved from https://mathworld.wolfram.com/LinearFunction.html
Glossary
- Linear function: A polynomial function of degree one, which means it has a single variable raised to the power of one.
- Slope-intercept form: A way of writing the equation of a line in terms of its slope and y-intercept.
- Slope: A measure of how steep a line is, calculated as the ratio of the vertical change (rise) to the horizontal change (run).
- Y-intercept: The point where the line intersects the y-axis.