The Graph Of Which Function Has An Axis Of Symmetry At $x = 37$?$f(x) = X^2 - 3x - 3$

Introduction

In mathematics, the axis of symmetry is a line that passes through the vertex of a parabola and divides it into two congruent halves. For a quadratic function in the form of $f(x) = ax^2 + bx + c$, the axis of symmetry can be found using the formula $x = -\frac{b}{2a}$. In this article, we will explore the graph of the function $f(x) = x^2 - 3x - 3$ and determine the axis of symmetry.

Understanding the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of a parabola. It is a key concept in graphing quadratic functions and is used to determine the shape and position of the graph. The axis of symmetry can be found using the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function.

Finding the Axis of Symmetry

To find the axis of symmetry of the function $f(x) = x^2 - 3x - 3$, we need to identify the values of $a$ and $b$. In this case, $a = 1$ and $b = -3$. Using the formula $x = -\frac{b}{2a}$, we can calculate the axis of symmetry as follows:

$x = -\frac{-3}{2(1)}$ $x = -\frac{-3}{2}$ $x = \frac{3}{2}$ $x = 1.5$

However, the problem statement gives us the axis of symmetry as $x = 37$. This suggests that the function $f(x) = x^2 - 3x - 3$ is not the correct function, or that there is an error in the problem statement.

Graphing the Function

To graph the function $f(x) = x^2 - 3x - 3$, we can start by identifying the vertex of the parabola. The vertex is the point on the graph where the axis of symmetry intersects the parabola. Since we are given the axis of symmetry as $x = 37$, we can use this information to find the vertex.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. To convert the function $f(x) = x^2 - 3x - 3$ to vertex form, we need to complete the square.

Completing the Square

To complete the square, we need to add and subtract $(\frac{b}{2})^2$ to the function. In this case, $b = -3$, so we need to add and subtract $(\frac{-3}{2})^2 = \frac{9}{4}$.

$f(x) = x^2 - 3x - 3$ $f(x) = (x^2 - 3x) - 3$ $f(x) = (x^2 - 3x + \frac{9}{4}) - 3 - \frac{9}{4}$ $f(x) = (x - \frac{3}{2})^2 - \frac{21}{4}$

Vertex Form

The vertex form of the function is $f(x) = (x - \frac{3}{2})^2 - \frac{21}{4}$. The vertex of the parabola is $(\frac{3}{2}, -\frac{21}{4})$.

Graphing the Vertex

To graph the vertex, we can use the coordinates $(\frac{3}{2}, -\frac{21}{4})$. The vertex is the point on the graph where the axis of symmetry intersects the parabola.

Conclusion

In this article, we explored the graph of the function $f(x) = x^2 - 3x - 3$ and determined the axis of symmetry. However, the problem statement gave us the axis of symmetry as $x = 37$, which suggests that the function $f(x) = x^2 - 3x - 3$ is not the correct function, or that there is an error in the problem statement. We also graphed the vertex of the parabola using the coordinates $(\frac{3}{2}, -\frac{21}{4})$.

References

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

Additional Resources

• [1] "Quadratic Functions" by Wolfram MathWorld
• [2] "Graphing Quadratic Functions" by Mathway
• [3] "Completing the Square" by MIT OpenCourseWare
  The Graph of a Quadratic Function with an Axis of Symmetry: Q&A ===========================================================

Introduction

In our previous article, we explored the graph of the function $f(x) = x^2 - 3x - 3$ and determined the axis of symmetry. However, the problem statement gave us the axis of symmetry as $x = 37$, which suggests that the function $f(x) = x^2 - 3x - 3$ is not the correct function, or that there is an error in the problem statement. In this article, we will answer some common questions related to the graph of a quadratic function with an axis of symmetry.

Q: What is the axis of symmetry of a quadratic function?

A: The axis of symmetry of a quadratic function is a vertical line that passes through the vertex of the parabola. It is a key concept in graphing quadratic functions and is used to determine the shape and position of the graph.

Q: How do I find the axis of symmetry of a quadratic function?

A: To find the axis of symmetry of a quadratic function, you need to identify the values of $a$ and $b$ in the function $f(x) = ax^2 + bx + c$. Then, you can use the formula $x = -\frac{b}{2a}$ to calculate the axis of symmetry.

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is given by $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. To convert the function $f(x) = x^2 - 3x - 3$ to vertex form, you need to complete the square.

Q: How do I complete the square to convert a quadratic function to vertex form?

A: To complete the square, you need to add and subtract $(\frac{b}{2})^2$ to the function. In this case, $b = -3$, so you need to add and subtract $(\frac{-3}{2})^2 = \frac{9}{4}$.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point on the graph where the axis of symmetry intersects the parabola. It is the minimum or maximum point of the graph, depending on the direction of the parabola.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to identify the vertex and the axis of symmetry. Then, you can use the coordinates of the vertex to graph the parabola.

Q: What are some common mistakes to avoid when graphing a quadratic function?

A: Some common mistakes to avoid when graphing a quadratic function include:

• Not identifying the vertex and axis of symmetry correctly
• Not completing the square correctly to convert the function to vertex form
• Not using the correct coordinates to graph the parabola

Conclusion

In this article, we answered some common questions related to the graph of a quadratic function with an axis of symmetry. We also provided some tips and tricks for graphing quadratic functions and avoiding common mistakes.

References

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

Additional Resources

• [1] "Quadratic Functions" by Wolfram MathWorld
• [2] "Graphing Quadratic Functions" by Mathway
• [3] "Completing the Square" by MIT OpenCourseWare