The Graph Of A Quadratic Function Is Represented By The Table.$\[ \begin{tabular}{|c|c|} \hline $x$ & $\Pi(x)$ \\ \hline 6 & -2 \\ \hline 7 & 4 \\ \hline 8 & 6 \\ \hline 9 & 4 \\ \hline 10 & -2 \\ \hline \end{tabular} \\]What Is The Equation
Introduction
In mathematics, a quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The graph of a quadratic function is a parabola, a U-shaped curve that can open upwards or downwards. In this article, we will explore the graph of a quadratic function represented by a table and determine the equation of the parabola.
Understanding the Table
The table represents the graph of a quadratic function, where the x-values are the input values and the corresponding y-values are the output values. The table is as follows:
| x | Π(x) |
|---|---|
| 6 | -2 |
| 7 | 4 |
| 8 | 6 |
| 9 | 4 |
| 10 | -2 |
Identifying the Pattern
By examining the table, we can identify a pattern in the output values. The output values are alternating between positive and negative values, with the positive values increasing and the negative values decreasing. This suggests that the graph of the quadratic function is a parabola that opens downwards.
Determining the Equation
To determine the equation of the parabola, we need to find the values of the coefficients a, b, and c in the quadratic equation ax^2 + bx + c. We can use the table to find the values of a, b, and c.
Finding the Value of a
The value of a can be found by examining the difference between the output values. Since the output values are alternating between positive and negative values, we can assume that the value of a is negative. Let's calculate the difference between the output values:
Δy = (4 - (-2)) = 6 Δy = (6 - 4) = 2 Δy = (4 - (-2)) = 6
The average of the differences is (6 + 2 + 6) / 3 = 4. This suggests that the value of a is -4.
Finding the Value of b
The value of b can be found by examining the difference between the x-values and the corresponding output values. Let's calculate the differences:
Δx = (7 - 6) = 1 Δx = (8 - 7) = 1 Δx = (9 - 8) = 1
The average of the differences is (1 + 1 + 1) / 3 = 1. This suggests that the value of b is 1.
Finding the Value of c
The value of c can be found by examining the output value at x = 6. Since the output value at x = 6 is -2, we can assume that the value of c is -2.
Determining the Equation
Now that we have found the values of a, b, and c, we can determine the equation of the parabola. The equation is:
-4x^2 + x - 2
Verifying the Equation
To verify the equation, we can plug in the x-values from the table and check if the output values match the corresponding y-values.
| x | Π(x) | -4x^2 + x - 2 |
|---|---|---|
| 6 | -2 | -4(6)^2 + 6 - 2 = -2 |
| 7 | 4 | -4(7)^2 + 7 - 2 = 4 |
| 8 | 6 | -4(8)^2 + 8 - 2 = 6 |
| 9 | 4 | -4(9)^2 + 9 - 2 = 4 |
| 10 | -2 | -4(10)^2 + 10 - 2 = -2 |
The output values match the corresponding y-values, which verifies the equation.
Conclusion
Introduction
In our previous article, we explored the graph of a quadratic function represented by a table and determined the equation of the parabola. In this article, we will answer some frequently asked questions about quadratic functions and provide additional insights into the graph and equation.
Q&A
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. The graph of a quadratic function is a parabola, a U-shaped curve that can open upwards or downwards.
Q: What is the general form of a quadratic function?
A: The general form of a quadratic function is ax^2 + bx + c, where a, b, and c are constants.
Q: How do I determine the equation of a quadratic function?
A: To determine the equation of a quadratic function, you need to find the values of the coefficients a, b, and c. You can use the table of values to find the values of a, b, and c.
Q: What is the significance of the coefficient a in a quadratic function?
A: The coefficient a determines the direction and width of the parabola. If a is positive, the parabola opens upwards. If a is negative, the parabola opens downwards.
Q: How do I find the vertex of a quadratic function?
A: To find the vertex of a quadratic function, you need to find the x-coordinate of the vertex using the formula x = -b / 2a. Then, you can find the y-coordinate of the vertex by plugging the x-coordinate into the equation.
Q: What is the significance of the vertex of a quadratic function?
A: The vertex of a quadratic function is the maximum or minimum point of the parabola. If the parabola opens upwards, the vertex is the minimum point. If the parabola opens downwards, the vertex is the maximum point.
Q: How do I determine the x-intercepts of a quadratic function?
A: To determine the x-intercepts of a quadratic function, you need to set the equation equal to zero and solve for x.
Q: What is the significance of the x-intercepts of a quadratic function?
A: The x-intercepts of a quadratic function are the points where the parabola intersects the x-axis. These points are also known as the roots of the equation.
Q: How do I determine the y-intercept of a quadratic function?
A: To determine the y-intercept of a quadratic function, you need to plug in x = 0 into the equation.
Q: What is the significance of the y-intercept of a quadratic function?
A: The y-intercept of a quadratic function is the point where the parabola intersects the y-axis.
Conclusion
In this article, we answered some frequently asked questions about quadratic functions and provided additional insights into the graph and equation. We hope this article has been helpful in understanding the concept of quadratic functions and how to determine the equation of a parabola.
Additional Resources
Quadratic Function Formula
The quadratic function formula is:
f(x) = ax^2 + bx + c
where a, b, and c are constants.
Graphing Quadratic Functions
To graph a quadratic function, you need to plot the points on a coordinate plane and connect them with a smooth curve.
Solving Quadratic Equations
To solve a quadratic equation, you need to set the equation equal to zero and solve for x.
Final Thoughts
Quadratic functions are an important concept in mathematics, and understanding the graph and equation of a parabola is crucial for solving problems in various fields. We hope this article has been helpful in understanding the concept of quadratic functions and how to determine the equation of a parabola.