The Tables Represent Two Linear Functions. The Equation Represented By The First Table Is Given Below: \[ \begin{tabular}{|c|c|} \hline X$ & Y Y Y \ \hline -4 & 11.5 \ \hline -3 & 17.25 \ \hline -2 & 23 \ \hline -1 & 28.75

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Introduction

In mathematics, linear functions are a fundamental concept that plays a crucial role in various mathematical operations. A linear function is a polynomial function of degree one, which means it has a single variable raised to the power of one. The general form of a linear function is f(x) = mx + b, where m is the slope and b is the y-intercept. In this article, we will discuss the equation represented by the first table and explore its properties.

The Equation Represented by the First Table

The equation represented by the first table is given below:

{ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline -4 & 11.5 \\ \hline -3 & 17.25 \\ \hline -2 & 23 \\ \hline -1 & 28.75 \end{tabular} }

To find the equation of the linear function represented by this table, we need to determine the slope (m) and the y-intercept (b). The slope can be calculated using the formula:

m = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are two points on the line.

Calculating the Slope

Using the points (-4, 11.5) and (-3, 17.25), we can calculate the slope as follows:

m = (17.25 - 11.5) / (-3 - (-4)) = 5.75 / 1 = 5.75

Calculating the Y-Intercept

Now that we have the slope, we can use any point on the line to calculate the y-intercept. Let's use the point (-4, 11.5):

b = y - mx = 11.5 - 5.75(-4) = 11.5 + 23 = 34.5

The Equation of the Linear Function

Now that we have the slope (m = 5.75) and the y-intercept (b = 34.5), we can write the equation of the linear function as follows:

f(x) = 5.75x + 34.5

Properties of the Linear Function

The linear function f(x) = 5.75x + 34.5 has several properties that are worth noting:

  • Domain: The domain of the linear function is all real numbers, which means that the function is defined for all values of x.
  • Range: The range of the linear function is all real numbers greater than or equal to 34.5, which means that the function takes on all values greater than or equal to 34.5.
  • Slope: The slope of the linear function is 5.75, which means that the function increases by 5.75 units for every 1 unit increase in x.
  • Y-Intercept: The y-intercept of the linear function is 34.5, which means that the function intersects the y-axis at the point (0, 34.5).

Graphing the Linear Function

To graph the linear function, we can use the slope and the y-intercept to plot two points on the line. Let's use the points (-4, 11.5) and (0, 34.5):

  • Plot the point (-4, 11.5) on the coordinate plane.
  • Plot the point (0, 34.5) on the coordinate plane.
  • Draw a line through the two points to form the graph of the linear function.

Conclusion

In this article, we discussed the equation represented by the first table and explored its properties. We calculated the slope and the y-intercept of the linear function and wrote the equation of the function in the form f(x) = mx + b. We also discussed the properties of the linear function, including its domain, range, slope, and y-intercept. Finally, we graphed the linear function using the slope and the y-intercept.

References

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: How do I calculate the slope of a linear function? A: You can calculate the slope using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.
  • Q: How do I calculate the y-intercept of a linear function? A: You can calculate the y-intercept using the formula b = y - mx, where m is the slope and (x, y) is a point on the line.

Glossary

  • Linear Function: A polynomial function of degree one, which means it has a single variable raised to the power of one.
  • Slope: The rate of change of a linear function, which is calculated using the formula m = (y2 - y1) / (x2 - x1).
  • Y-Intercept: The point at which a linear function intersects the y-axis, which is calculated using the formula b = y - mx.
    The Tables Represent Two Linear Functions: Understanding the Equation - Q&A ====================================================================

Introduction

In our previous article, we discussed the equation represented by the first table and explored its properties. We calculated the slope and the y-intercept of the linear function and wrote the equation of the function in the form f(x) = mx + b. We also discussed the properties of the linear function, including its domain, range, slope, and y-intercept. In this article, we will answer some frequently asked questions about linear functions.

Q&A

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. The general form of a linear function is f(x) = mx + b, where m is the slope and b is the y-intercept.

Q: How do I calculate the slope of a linear function?

A: You can calculate the slope using the formula m = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are two points on the line.

Q: How do I calculate the y-intercept of a linear function?

A: You can calculate the y-intercept using the formula b = y - mx, where m is the slope and (x, y) is a point on the line.

Q: What is the domain of a linear function?

A: The domain of a linear function is all real numbers, which means that the function is defined for all values of x.

Q: What is the range of a linear function?

A: The range of a linear function is all real numbers greater than or equal to the y-intercept, which means that the function takes on all values greater than or equal to the y-intercept.

Q: How do I graph a linear function?

A: To graph a linear function, you can use the slope and the y-intercept to plot two points on the line. Then, draw a line through the two points to form the graph of the linear function.

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. The general form of a linear function is f(x) = mx + b, while the general form of a quadratic function is f(x) = ax^2 + bx + c.

Q: Can a linear function have a negative slope?

A: Yes, a linear function can have a negative slope. If the slope is negative, the function will decrease as x increases.

Q: Can a linear function have a zero slope?

A: Yes, a linear function can have a zero slope. If the slope is zero, the function will be a horizontal line.

Q: Can a linear function have a fractional slope?

A: Yes, a linear function can have a fractional slope. If the slope is a fraction, the function will have a non-integer rate of change.

Q: Can a linear function have a negative y-intercept?

A: Yes, a linear function can have a negative y-intercept. If the y-intercept is negative, the function will intersect the y-axis at a point below the origin.

Q: Can a linear function have a fractional y-intercept?

A: Yes, a linear function can have a fractional y-intercept. If the y-intercept is a fraction, the function will intersect the y-axis at a non-integer point.

Conclusion

In this article, we answered some frequently asked questions about linear functions. We discussed the definition of a linear function, how to calculate the slope and y-intercept, and how to graph a linear function. We also compared linear functions to quadratic functions and discussed some special cases of linear functions.

References

Glossary

  • Linear Function: A polynomial function of degree one, which means it has a single variable raised to the power of one.
  • Slope: The rate of change of a linear function, which is calculated using the formula m = (y2 - y1) / (x2 - x1).
  • Y-Intercept: The point at which a linear function intersects the y-axis, which is calculated using the formula b = y - mx.
  • Domain: The set of all possible input values for a function.
  • Range: The set of all possible output values for a function.
  • Quadratic Function: A polynomial function of degree two, which means it has a single variable raised to the power of two.