Complete The Table For The Function $f(a)=9a^2+9a$.${ \begin{array}{|c|c|} \hline a & F(a) \ \hline -3 & 1 \ \hline -2 & \ \hline -1 & \ \hline 0 & I \ \hline \end{array} }$
Understanding the Function
The given function is a quadratic function in the form of f(a) = 9a^2 + 9a. This function represents a parabola that opens upwards, and its vertex can be found using the formula a = -b / 2a, where a and b are coefficients of the quadratic function.
Calculating f(a) for Different Values of a
To complete the table, we need to calculate f(a) for different values of a. We are given the values of a as -3, -2, -1, and 0. We can calculate f(a) for these values using the given function.
Calculating f(-3)
To calculate f(-3), we substitute a = -3 into the function f(a) = 9a^2 + 9a.
f(-3) = 9(-3)^2 + 9(-3) f(-3) = 9(9) - 27 f(-3) = 81 - 27 f(-3) = 54
Calculating f(-2)
To calculate f(-2), we substitute a = -2 into the function f(a) = 9a^2 + 9a.
f(-2) = 9(-2)^2 + 9(-2) f(-2) = 9(4) - 18 f(-2) = 36 - 18 f(-2) = 18
Calculating f(-1)
To calculate f(-1), we substitute a = -1 into the function f(a) = 9a^2 + 9a.
f(-1) = 9(-1)^2 + 9(-1) f(-1) = 9(1) - 9 f(-1) = 9 - 9 f(-1) = 0
Calculating f(0)
To calculate f(0), we substitute a = 0 into the function f(a) = 9a^2 + 9a.
f(0) = 9(0)^2 + 9(0) f(0) = 0 + 0 f(0) = 0
Completing the Table
Now that we have calculated f(a) for different values of a, we can complete the table.
| a | f(a) |
|---|---|
| -3 | 54 |
| -2 | 18 |
| -1 | 0 |
| 0 | 0 |
Discussion
The completed table shows the values of f(a) for different values of a. We can see that the function f(a) = 9a^2 + 9a is a quadratic function that opens upwards. The vertex of the parabola can be found using the formula a = -b / 2a, where a and b are coefficients of the quadratic function.
Conclusion
In this article, we completed the table for the function f(a) = 9a^2 + 9a. We calculated f(a) for different values of a and completed the table. We also discussed the properties of the quadratic function and its vertex.
Future Work
In the future, we can explore other properties of the quadratic function, such as its axis of symmetry and its x-intercepts. We can also investigate the behavior of the function as a approaches positive or negative infinity.
References
- [1] "Quadratic Functions" by Math Open Reference
- [2] "Vertex Form of a Quadratic Function" by Purplemath
Glossary
- Quadratic function: A polynomial function of degree two, which can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants.
- Vertex: The point on the graph of a quadratic function where the function changes direction.
- Axis of symmetry: The line that passes through the vertex of a quadratic function and is perpendicular to the x-axis.
- X-intercept: The point on the graph of a quadratic function where the function intersects the x-axis.
Understanding the Function
The function f(a) = 9a^2 + 9a is a quadratic function that represents a parabola that opens upwards. In this article, we will answer some frequently asked questions about this function.
Q: What is the vertex of the parabola represented by the function f(a) = 9a^2 + 9a?
A: The vertex of the parabola can be found using the formula a = -b / 2a, where a and b are coefficients of the quadratic function. In this case, a = 9 and b = 9, so the vertex is a = -9 / (2*9) = -1/2.
Q: What is the axis of symmetry of the parabola represented by the function f(a) = 9a^2 + 9a?
A: The axis of symmetry is the line that passes through the vertex of the parabola and is perpendicular to the x-axis. Since the vertex is at a = -1/2, the axis of symmetry is the vertical line x = -1/2.
Q: What are the x-intercepts of the parabola represented by the function f(a) = 9a^2 + 9a?
A: The x-intercepts are the points on the graph of the function where the function intersects the x-axis. To find the x-intercepts, we set f(a) = 0 and solve for a. In this case, we have 9a^2 + 9a = 0, which can be factored as 9a(a + 1) = 0. This gives us two possible values for a: a = 0 and a = -1.
Q: What is the behavior of the function f(a) = 9a^2 + 9a as a approaches positive or negative infinity?
A: As a approaches positive or negative infinity, the function f(a) = 9a^2 + 9a approaches positive infinity. This is because the quadratic term 9a^2 dominates the linear term 9a as a becomes large.
Q: Can the function f(a) = 9a^2 + 9a be written in vertex form?
A: Yes, the function f(a) = 9a^2 + 9a can be written in vertex form as f(a) = 9(a + 1/2)^2 - 9/4.
Q: What is the domain of the function f(a) = 9a^2 + 9a?
A: The domain of the function f(a) = 9a^2 + 9a is all real numbers, since the function is defined for all values of a.
Q: What is the range of the function f(a) = 9a^2 + 9a?
A: The range of the function f(a) = 9a^2 + 9a is all non-negative real numbers, since the function is always non-negative.
Conclusion
In this article, we answered some frequently asked questions about the function f(a) = 9a^2 + 9a. We discussed the vertex, axis of symmetry, x-intercepts, behavior as a approaches positive or negative infinity, vertex form, domain, and range of the function.
References
- [1] "Quadratic Functions" by Math Open Reference
- [2] "Vertex Form of a Quadratic Function" by Purplemath
Glossary
- Quadratic function: A polynomial function of degree two, which can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants.
- Vertex: The point on the graph of a quadratic function where the function changes direction.
- Axis of symmetry: The line that passes through the vertex of a quadratic function and is perpendicular to the x-axis.
- X-intercept: The point on the graph of a quadratic function where the function intersects the x-axis.
- Domain: The set of all possible input values for a function.
- Range: The set of all possible output values for a function.