{ \begin{tabular}{|c|c|} \hline $x$ & $f(x)$ \\ \hline -2 & 21 \\ \hline -1 & 10 \\ \hline 0 & 5 \\ \hline 1 & 6 \\ \hline 2 & 13 \\ \hline \end{tabular} \}$Which Quadratic Function Is Represented By The Table?A. $f(x) = 3x^2 + 2x -
Introduction
In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. These functions are commonly represented in the form of f(x) = ax^2 + bx + c, where a, b, and c are constants. In this article, we will explore how to determine the quadratic function represented by a given table of values.
Understanding the Table
The table provided contains six data points, each with a corresponding x-value and f(x) value. The x-values range from -2 to 2, and the f(x) values range from 5 to 21. To determine the quadratic function represented by this table, we need to analyze the pattern of the data points and find a function that fits all the given values.
Analyzing the Data Points
Let's examine the data points in the table:
| x | f(x) |
|---|---|
| -2 | 21 |
| -1 | 10 |
| 0 | 5 |
| 1 | 6 |
| 2 | 13 |
From the table, we can observe that the f(x) values are not in a simple arithmetic or geometric sequence. However, we can try to find a pattern by examining the differences between consecutive f(x) values.
Finding the Pattern
Let's calculate the differences between consecutive f(x) values:
| x | f(x) | Δf(x) |
|---|---|---|
| -2 | 21 | - |
| -1 | 10 | -11 |
| 0 | 5 | -5 |
| 1 | 6 | 1 |
| 2 | 13 | 7 |
From the table, we can see that the differences between consecutive f(x) values are not constant. However, we can try to find a pattern by examining the second differences.
Calculating the Second Differences
Let's calculate the second differences between consecutive f(x) values:
| x | f(x) | Δf(x) | Δ²f(x) |
|---|---|---|---|
| -2 | 21 | -11 | - |
| -1 | 10 | -5 | 6 |
| 0 | 5 | 1 | -5 |
| 1 | 6 | 7 | 6 |
| 2 | 13 | - | - |
From the table, we can see that the second differences are constant, with a value of 6. This suggests that the quadratic function represented by the table is a quadratic function with a constant second derivative.
Determining the Quadratic Function
Based on the analysis of the data points and the second differences, we can determine the quadratic function represented by the table. Since the second differences are constant, we can write the quadratic function in the form of f(x) = ax^2 + bx + c, where a is the coefficient of the x^2 term, b is the coefficient of the x term, and c is the constant term.
To determine the values of a, b, and c, we can use the data points in the table. We can start by using the first data point (x = -2, f(x) = 21) to find the value of a. We can then use the second data point (x = -1, f(x) = 10) to find the value of b. Finally, we can use the third data point (x = 0, f(x) = 5) to find the value of c.
Solving for a, b, and c
Let's start by using the first data point (x = -2, f(x) = 21) to find the value of a. We can plug in the values of x and f(x) into the quadratic function f(x) = ax^2 + bx + c and solve for a:
21 = a(-2)^2 + b(-2) + c 21 = 4a - 2b + c
Next, let's use the second data point (x = -1, f(x) = 10) to find the value of b. We can plug in the values of x and f(x) into the quadratic function f(x) = ax^2 + bx + c and solve for b:
10 = a(-1)^2 + b(-1) + c 10 = a - b + c
Finally, let's use the third data point (x = 0, f(x) = 5) to find the value of c. We can plug in the values of x and f(x) into the quadratic function f(x) = ax^2 + bx + c and solve for c:
5 = a(0)^2 + b(0) + c 5 = c
Now that we have found the value of c, we can substitute it into the previous two equations to solve for a and b:
21 = 4a - 2b + 5 10 = a - b + 5
Simplifying the equations, we get:
16 = 4a - 2b 5 = a - b
We can solve this system of equations by multiplying the second equation by 2 and adding it to the first equation:
16 = 4a - 2b 10 = 2a - 2b 26 = 6a 4.33 = a
Now that we have found the value of a, we can substitute it into one of the previous equations to solve for b. Let's use the second equation:
5 = a - b 5 = 4.33 - b b = -0.33
Now that we have found the values of a, b, and c, we can write the quadratic function represented by the table:
f(x) = 4.33x^2 - 0.33x + 5
Conclusion
In this article, we have determined the quadratic function represented by a given table of values. We analyzed the data points in the table and found a pattern by examining the differences between consecutive f(x) values. We then calculated the second differences and found that they were constant, which suggested that the quadratic function represented by the table was a quadratic function with a constant second derivative. We used the data points in the table to solve for the values of a, b, and c, and found that the quadratic function represented by the table was f(x) = 4.33x^2 - 0.33x + 5.
Discussion
The quadratic function represented by the table is a quadratic function with a positive leading coefficient (a = 4.33). This means that the parabola represented by the function opens upward. The vertex of the parabola is located at the point (h, k) = (-b/2a, f(h)), where h = -b/2a and k = f(h). In this case, h = -(-0.33)/(2*4.33) = 0.038 and k = f(0.038) = 5.001. Therefore, the vertex of the parabola is located at the point (0.038, 5.001).
The quadratic function represented by the table has a minimum value of 5.001, which occurs at the vertex of the parabola. The function has a maximum value of 21, which occurs at the point (x, f(x)) = (-2, 21).
Final Answer
Introduction
In our previous article, we determined the quadratic function represented by a given table of values. In this article, we will answer some frequently asked questions about quadratic functions and provide additional information to help you better understand these functions.
Q: What is a quadratic function?
A: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. These functions are commonly represented in the form of f(x) = ax^2 + bx + c, where a, b, and c are constants.
Q: What are the characteristics of a quadratic function?
A: Quadratic functions have several characteristics, including:
- A parabolic shape: Quadratic functions have a parabolic shape, which means they have a single turning point, called the vertex.
- A minimum or maximum value: Quadratic functions have a minimum or maximum value, depending on the sign of the leading coefficient (a).
- A single root: Quadratic functions have a single root, which is the value of x that makes the function equal to zero.
Q: How do I determine the vertex of a quadratic function?
A: To determine the vertex of a quadratic function, you can use the formula h = -b/2a, where h is the x-coordinate of the vertex and a and b are the coefficients of the quadratic function.
Q: How do I determine the minimum or maximum value of a quadratic function?
A: To determine the minimum or maximum value of a quadratic function, you can use the formula k = f(h), where k is the y-coordinate of the vertex and h is the x-coordinate of the vertex.
Q: What is the difference between a quadratic function and a linear function?
A: A quadratic function is a polynomial function of degree two, while a linear function is a polynomial function of degree one. Quadratic functions have a parabolic shape, while linear functions have a straight line shape.
Q: Can I use a quadratic function to model real-world phenomena?
A: Yes, quadratic functions can be used to model real-world phenomena, such as the trajectory of a projectile, the motion of an object under the influence of gravity, or the growth of a population.
Q: How do I graph a quadratic function?
A: To graph a quadratic function, you can use a graphing calculator or a computer program to plot the function. You can also use a table of values to plot the function.
Q: Can I use a quadratic function to solve a system of equations?
A: Yes, quadratic functions can be used to solve a system of equations. You can use the quadratic formula to solve for the roots of the quadratic function, and then use the roots to solve for the values of the variables in the system of equations.
Q: What are some common applications of quadratic functions?
A: Quadratic functions have many common applications, including:
- Physics: Quadratic functions are used to model the motion of objects under the influence of gravity.
- Engineering: Quadratic functions are used to design and optimize systems, such as bridges and buildings.
- Economics: Quadratic functions are used to model the behavior of economic systems, such as supply and demand curves.
- Computer Science: Quadratic functions are used to solve problems in computer science, such as finding the shortest path between two points.
Conclusion
In this article, we have answered some frequently asked questions about quadratic functions and provided additional information to help you better understand these functions. Quadratic functions are an important tool in mathematics and have many common applications in physics, engineering, economics, and computer science.