If $f(x)$ Is A Linear Function, What Is The Value Of $n$?$\[ \begin{tabular}{|c|c|} \hline $x$ & $f(x)$ \\ \hline -4 & -25 \\ \hline -1 & -10 \\ \hline $n$ & 20 \\ \hline \end{tabular} \\]A. 2 B. 4 C. 5 D. 9

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Introduction

In mathematics, a linear function is a polynomial function of degree one, which means it can be written in the form of f(x)=mx+bf(x) = mx + b, where mm is the slope and bb is the y-intercept. In this article, we will explore how to find the value of nn in a linear function given a table of values.

Understanding Linear Functions

A linear function is a function that can be written in the form of f(x)=mx+bf(x) = mx + b. The slope mm represents the rate of change of the function, and the y-intercept bb represents the point where the function intersects the y-axis.

The Table of Values

The table of values provided shows the input xx and the corresponding output f(x)f(x) for three different values of xx. The table is as follows:

xx f(x)f(x)
-4 -25
-1 -10
nn 20

Finding the Slope

To find the value of nn, we need to first find the slope of the linear function. We can do this by using the two points in the table, (-4, -25) and (-1, -10).

The slope mm can be calculated using the formula:

m=f(x2)−f(x1)x2−x1m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}

Substituting the values from the table, we get:

m=−10−(−25)−1−(−4)m = \frac{-10 - (-25)}{-1 - (-4)}

m=153m = \frac{15}{3}

m=5m = 5

Finding the Y-Intercept

Now that we have the slope, we can find the y-intercept bb by using one of the points in the table. Let's use the point (-4, -25).

We can substitute the values into the equation f(x)=mx+bf(x) = mx + b:

−25=5(−4)+b-25 = 5(-4) + b

−25=−20+b-25 = -20 + b

b=−5b = -5

Finding the Value of n

Now that we have the slope and the y-intercept, we can find the value of nn by using the equation f(x)=mx+bf(x) = mx + b.

We are given that f(n)=20f(n) = 20, so we can substitute this value into the equation:

20=5n−520 = 5n - 5

25=5n25 = 5n

n=5n = 5

Conclusion

In this article, we explored how to find the value of nn in a linear function given a table of values. We first found the slope of the linear function by using the two points in the table, and then we found the y-intercept by using one of the points. Finally, we used the equation f(x)=mx+bf(x) = mx + b to find the value of nn.

The final answer is 5\boxed{5}.

Discussion

This problem is a great example of how to use linear functions to solve real-world problems. In this case, we were given a table of values and asked to find the value of nn. By using the slope and y-intercept, we were able to find the value of nn.

Related Topics

  • Linear functions
  • Slope and y-intercept
  • Tables of values
  • Real-world applications of linear functions

References

Q: What is a linear function?

A: A linear function is a polynomial function of degree one, which means it can be written in the form of f(x)=mx+bf(x) = mx + b, where mm is the slope and bb is the y-intercept.

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

A: To find the slope of a linear function, you can use the formula:

m=f(x2)−f(x1)x2−x1m = \frac{f(x_2) - f(x_1)}{x_2 - x_1}

This formula uses two points on the graph of the function to calculate the slope.

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

A: To find the y-intercept of a linear function, you can use the equation f(x)=mx+bf(x) = mx + b and substitute the values of mm and xx into the equation. The y-intercept is the value of bb.

Q: How do I use the table of values to find the value of n?

A: To use the table of values to find the value of nn, you can first find the slope of the linear function by using the two points in the table. Then, you can use the equation f(x)=mx+bf(x) = mx + b to find the value of nn.

Q: What if I don't have a table of values? How can I find the value of n?

A: If you don't have a table of values, you can use the equation f(x)=mx+bf(x) = mx + b and substitute the values of mm and xx into the equation to find the value of nn.

Q: Can I use the slope and y-intercept to find the value of n?

A: Yes, you can use the slope and y-intercept to find the value of nn. By substituting the values of mm and bb into the equation f(x)=mx+bf(x) = mx + b, you can solve for nn.

Q: What if I get a negative value for n? Is that possible?

A: Yes, it is possible to get a negative value for nn. However, in this problem, we are looking for a positive value for nn.

Q: Can I use this method to find the value of n for any linear function?

A: Yes, this method can be used to find the value of nn for any linear function.

Q: Are there any other ways to find the value of n?

A: Yes, there are other ways to find the value of nn. For example, you can use the equation f(x)=mx+bf(x) = mx + b and substitute the values of mm and xx into the equation to solve for nn.

Q: Can I use this method to find the value of n for a quadratic function?

A: No, this method is only for linear functions. For quadratic functions, you would need to use a different method to find the value of nn.

Q: Are there any online resources that can help me with this problem?

A: Yes, there are many online resources that can help you with this problem. You can try searching for "linear functions" or "slope and y-intercept" on a search engine to find helpful resources.

Q: Can I use this method to find the value of n for a function with multiple variables?

A: No, this method is only for functions with one variable. For functions with multiple variables, you would need to use a different method to find the value of nn.

Q: Are there any books or textbooks that can help me with this problem?

A: Yes, there are many books and textbooks that can help you with this problem. You can try searching for "linear functions" or "slope and y-intercept" in a bookstore or online to find helpful resources.

Q: Can I use this method to find the value of n for a function with a non-linear relationship?

A: No, this method is only for linear functions. For functions with a non-linear relationship, you would need to use a different method to find the value of nn.