12. \[$ F(x) = X^2 - 81 \$\]- Vertex: - Minimum Or Maximum: - Zeroes: - Axis Of Symmetry (AoS): - Y-intercept:

Understanding the Graph of a Quadratic Function: A Comprehensive Analysis of $f(x) = x^2 - 81$

Quadratic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will delve into the graph of the quadratic function $f(x) = x^2 - 81$ and analyze its key features, including the vertex, minimum or maximum, zeroes, axis of symmetry, and y-intercept.

Vertex

The vertex of a quadratic function is the point at which the function changes from decreasing to increasing or vice versa. To find the vertex of the function $f(x) = x^2 - 81$, we can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. In this case, $a = 1$ and $b = 0$, so the x-coordinate of the vertex is $x = -\frac{0}{2(1)} = 0$. To find the y-coordinate of the vertex, we substitute $x = 0$ into the function: $f(0) = (0)^2 - 81 = -81$. Therefore, the vertex of the function $f(x) = x^2 - 81$ is $(0, -81)$.

Minimum or Maximum

Since the coefficient of the squared term is positive ($a = 1 > 0$), the function $f(x) = x^2 - 81$ is a concave-up function, and its vertex represents the minimum point of the function. This means that the function decreases on both sides of the vertex and increases as we move away from the vertex.

Zeroes

To find the zeroes of the function $f(x) = x^2 - 81$, we set the function equal to zero and solve for $x$: $x^2 - 81 = 0$. Adding $81$ to both sides gives us $x^2 = 81$. Taking the square root of both sides, we get $x = \pm \sqrt{81} = \pm 9$. Therefore, the zeroes of the function $f(x) = x^2 - 81$ are $x = 9$ and $x = -9$.

Axis of Symmetry (AoS)

The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the function. Since the vertex of the function $f(x) = x^2 - 81$ is $(0, -81)$, the axis of symmetry is the vertical line $x = 0$.

y-intercept

The y-intercept of a function is the point at which the function intersects the y-axis. To find the y-intercept of the function $f(x) = x^2 - 81$, we substitute $x = 0$ into the function: $f(0) = (0)^2 - 81 = -81$. Therefore, the y-intercept of the function $f(x) = x^2 - 81$ is $(0, -81)$.

Graph of the Function

The graph of the function $f(x) = x^2 - 81$ is a parabola that opens upwards, with its vertex at $(0, -81)$. The graph has two zeroes at $x = 9$ and $x = -9$, and its axis of symmetry is the vertical line $x = 0$. The graph also passes through the point $(0, -81)$, which is the y-intercept of the function.

Real-World Applications

Quadratic functions have numerous real-world applications, including physics, engineering, and economics. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic function. Similarly, the cost of producing a certain quantity of goods can be represented by a quadratic function.

Conclusion

In conclusion, the graph of the quadratic function $f(x) = x^2 - 81$ has several key features, including a vertex at $(0, -81)$, a minimum point, zeroes at $x = 9$ and $x = -9$, an axis of symmetry at $x = 0$, and a y-intercept at $(0, -81)$. Understanding these features is essential for solving various mathematical problems and has numerous real-world applications.

References

• [1] "Quadratic Functions" by Math Open Reference
• [2] "Graphing Quadratic Functions" by Purplemath
• [3] "Quadratic Equations" by Khan Academy

Further Reading

• "Quadratic Functions: A Comprehensive Guide" by Math Is Fun
• "Graphing Quadratic Functions: A Step-by-Step Guide" by IXL
• "Quadratic Equations: A Tutorial" by MIT OpenCourseWare
  Quadratic Function $f(x) = x^2 - 81$: A Q&A Guide

In our previous article, we explored the graph of the quadratic function $f(x) = x^2 - 81$ and analyzed its key features, including the vertex, minimum or maximum, zeroes, axis of symmetry, and y-intercept. In this article, we will answer some frequently asked questions about the quadratic function $f(x) = x^2 - 81$.

Q: What is the vertex of the function $f(x) = x^2 - 81$?

A: The vertex of the function $f(x) = x^2 - 81$ is the point at which the function changes from decreasing to increasing or vice versa. To find the vertex, we can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. In this case, $a = 1$ and $b = 0$, so the x-coordinate of the vertex is $x = -\frac{0}{2(1)} = 0$. To find the y-coordinate of the vertex, we substitute $x = 0$ into the function: $f(0) = (0)^2 - 81 = -81$. Therefore, the vertex of the function $f(x) = x^2 - 81$ is $(0, -81)$.

Q: Is the function $f(x) = x^2 - 81$ a minimum or maximum?

A: Since the coefficient of the squared term is positive ($a = 1 > 0$), the function $f(x) = x^2 - 81$ is a concave-up function, and its vertex represents the minimum point of the function. This means that the function decreases on both sides of the vertex and increases as we move away from the vertex.

Q: What are the zeroes of the function $f(x) = x^2 - 81$?

A: To find the zeroes of the function $f(x) = x^2 - 81$, we set the function equal to zero and solve for $x$: $x^2 - 81 = 0$. Adding $81$ to both sides gives us $x^2 = 81$. Taking the square root of both sides, we get $x = \pm \sqrt{81} = \pm 9$. Therefore, the zeroes of the function $f(x) = x^2 - 81$ are $x = 9$ and $x = -9$.

Q: What is the axis of symmetry of the function $f(x) = x^2 - 81$?

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the function. Since the vertex of the function $f(x) = x^2 - 81$ is $(0, -81)$, the axis of symmetry is the vertical line $x = 0$.

Q: What is the y-intercept of the function $f(x) = x^2 - 81$?

A: The y-intercept of a function is the point at which the function intersects the y-axis. To find the y-intercept of the function $f(x) = x^2 - 81$, we substitute $x = 0$ into the function: $f(0) = (0)^2 - 81 = -81$. Therefore, the y-intercept of the function $f(x) = x^2 - 81$ is $(0, -81)$.

Q: How can I use the quadratic function $f(x) = x^2 - 81$ in real-world applications?

A: Quadratic functions have numerous real-world applications, including physics, engineering, and economics. For example, the trajectory of a projectile under the influence of gravity can be modeled using a quadratic function. Similarly, the cost of producing a certain quantity of goods can be represented by a quadratic function.

Q: Can I use the quadratic function $f(x) = x^2 - 81$ to model a situation where the cost increases as the quantity produced increases?

A: No, the quadratic function $f(x) = x^2 - 81$ is a concave-up function, which means that it decreases as the input increases. This is not suitable for modeling a situation where the cost increases as the quantity produced increases.

Q: Can I use the quadratic function $f(x) = x^2 - 81$ to model a situation where the cost decreases as the quantity produced increases?

A: Yes, the quadratic function $f(x) = x^2 - 81$ is a concave-up function, which means that it decreases as the input increases. This is suitable for modeling a situation where the cost decreases as the quantity produced increases.

Q: How can I graph the quadratic function $f(x) = x^2 - 81$?

A: To graph the quadratic function $f(x) = x^2 - 81$, you can use a graphing calculator or a computer algebra system. Alternatively, you can plot the function by hand using a table of values.

Q: Can I use the quadratic function $f(x) = x^2 - 81$ to model a situation where the quantity produced is a function of time?

A: Yes, the quadratic function $f(x) = x^2 - 81$ can be used to model a situation where the quantity produced is a function of time. However, you would need to modify the function to include a time variable.

Q: Can I use the quadratic function $f(x) = x^2 - 81$ to model a situation where the cost is a function of the quantity produced and time?

A: Yes, the quadratic function $f(x) = x^2 - 81$ can be used to model a situation where the cost is a function of the quantity produced and time. However, you would need to modify the function to include both the quantity produced and time variables.

Conclusion

In conclusion, the quadratic function $f(x) = x^2 - 81$ has several key features, including a vertex at $(0, -81)$, a minimum point, zeroes at $x = 9$ and $x = -9$, an axis of symmetry at $x = 0$, and a y-intercept at $(0, -81)$. Understanding these features is essential for solving various mathematical problems and has numerous real-world applications.