Plotting the Axis of Symmetry

Oct 13, 2025 by talynt.org 113 views

Introduction

In mathematics, functions are used to describe the relationship between variables. A function can be represented graphically on a coordinate plane, which provides a visual representation of the function's behavior. In this article, we will consider the function $f(x) = x^2 - 6x + 5$ and plot its key features on the provided coordinate plane.

The Function

The given function is a quadratic function, which can be written in the form $f(x) = ax^2 + bx + c$ . In this case, the function is $f(x) = x^2 - 6x + 5$ , where $a = 1$ , $b = -6$ , and $c = 5$ .

Plotting the x-intercept(s)

To plot the x-intercept(s), we need to find the value(s) of x that make the function equal to zero. This can be done by setting $f(x) = 0$ and solving for x.

import sympy as sp
x = sp.symbols('x')
f = x**2 - 6*x + 5
solutions = sp.solve(f, x)
print(solutions)

The output of the above code will be the x-intercept(s) of the function.

Plotting the y-intercept

To plot the y-intercept, we need to find the value of y when x is equal to zero. This can be done by substituting x = 0 into the function.

# Substitute x = 0 into the function
y_intercept = f.subs(x, 0)
print(y_intercept)

The output of the above code will be the y-intercept of the function.

Plotting the Vertex

The vertex of a quadratic function is the point on the graph where the function changes from decreasing to increasing or vice versa. The x-coordinate of the vertex can be found using the formula $x = -\frac{b}{2a}$ .

# Calculate the x-coordinate of the vertex
vertex_x = -(-6) / (2 * 1)
print(vertex_x)

The y-coordinate of the vertex can be found by substituting the x-coordinate into the function.

# Calculate the y-coordinate of the vertex
vertex_y = f.subs(x, vertex_x)
print(vertex_y)

Plotting the Axis of Symmetry

The axis of symmetry of a quadratic function is the vertical line that passes through the vertex. The equation of the axis of symmetry can be found using the formula $x = -\frac{b}{2a}$ .

# Calculate the equation of the axis of symmetry
axis_of_symmetry = x - vertex_x
print(axis_of_symmetry)

Conclusion

In this article, we have considered the function $f(x) = x^2 - 6x + 5$ and plotted its key features on the provided coordinate plane. We have found the x-intercept(s), y-intercept, vertex, and axis of symmetry of the function using mathematical techniques and Python code. The graphical representation of the function provides a visual understanding of its behavior and can be used to make predictions and draw conclusions about the function's properties.

Key Takeaways

The x-intercept(s) of a quadratic function can be found by setting the function equal to zero and solving for x.
The y-intercept of a quadratic function can be found by substituting x = 0 into the function.
The vertex of a quadratic function can be found using the formula $x = -\frac{b}{2a}$ .
The axis of symmetry of a quadratic function can be found using the formula $x = -\frac{b}{2a}$ .

References

[1] Sympy Documentation. (n.d.). Retrieved from https://docs.sympy.org/latest/index.html
[2] Khan Academy. (n.d.). Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f4f7d0/x2f4f7d0/quadratic-functions

Future Work

Investigate the properties of the function $f(x) = x^2 - 6x + 5$ in more detail, such as its domain and range.
Explore the relationship between the function and its graphical representation.
Use the function to model real-world phenomena and make predictions about the behavior of the system.

Code

import sympy as sp
x = sp.symbols('x')
f = x**2 - 6*x + 5
solutions = sp.solve(f, x)
print(solutions)
y_intercept = f.subs(x, 0)
print(y_intercept)
vertex_x = -(-6) / (2 * 1)
print(vertex_x)
vertex_y = f.subs(x, vertex_x)
print(vertex_y)
axis_of_symmetry = x - vertex_x
print(axis_of_symmetry)
**Quadratic Function Q&amp;A**
==========================

### Frequently Asked Questions

#### Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means it has a highest power of two. It can be written in the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

#### Q: What are the key features of a quadratic function?

A: The key features of a quadratic function are:

* The x-intercept(s): The points on the graph where the function crosses the x-axis.
* The y-intercept: The point on the graph where the function crosses the y-axis.
* The vertex: The point on the graph where the function changes from decreasing to increasing or vice versa.
* The axis of symmetry: The vertical line that passes through the vertex.

#### Q: How do I find the x-intercept(s) of a quadratic function?

A: To find the x-intercept(s) of a quadratic function, you need to set the function equal to zero and solve for x. This can be done using algebraic techniques or numerical methods.

#### Q: How do I find the y-intercept of a quadratic function?

A: To find the y-intercept of a quadratic function, you need to substitute x = 0 into the function. This will give you the value of y when x is equal to zero.

#### Q: How do I find the vertex of a quadratic function?

A: To find the vertex of a quadratic function, you need to use the formula $x = -\frac{b}{2a}$. This will give you the x-coordinate of the vertex. You can then substitute this value into the function to find the y-coordinate of the vertex.

#### 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 use the formula $x = -\frac{b}{2a}$. This will give you the equation of the axis of symmetry.

#### 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. This means that a quadratic function has a highest power of two, while a linear function has a highest power of one.

#### Q: Can a quadratic function have more than one x-intercept?

A: Yes, a quadratic function can have more than one x-intercept. This occurs when the function has two real roots.

#### Q: Can a quadratic function have no x-intercepts?

A: Yes, a quadratic function can have no x-intercepts. This occurs when the function has no real roots.

#### Q: Can a quadratic function have a negative leading coefficient?

A: Yes, a quadratic function can have a negative leading coefficient. This means that the function will have a downward opening parabola.

#### Q: Can a quadratic function have a positive leading coefficient?

A: Yes, a quadratic function can have a positive leading coefficient. This means that the function will have an upward opening parabola.

### Conclusion

In this article, we have answered some of the most frequently asked questions about quadratic functions. We have covered topics such as the key features of a quadratic function, how to find the x-intercept(s), y-intercept, vertex, and axis of symmetry, and the difference between a quadratic function and a linear function. We hope that this article has been helpful in answering your questions about quadratic functions.

### Key Takeaways

* A quadratic function is a polynomial function of degree two.
* The key features of a quadratic function are the x-intercept(s), y-intercept, vertex, and axis of symmetry.
* To find the x-intercept(s) of a quadratic function, you need to set the function equal to zero and solve for x.
* To find the y-intercept of a quadratic function, you need to substitute x = 0 into the function.
* To find the vertex of a quadratic function, you need to use the formula $x = -\frac{b}{2a}$.
* To find the axis of symmetry of a quadratic function, you need to use the formula $x = -\frac{b}{2a}$.

### References

* [1] Sympy Documentation. (n.d.). Retrieved from &lt;https://docs.sympy.org/latest/index.html&gt;
* [2] Khan Academy. (n.d.). Quadratic Functions. Retrieved from &lt;https://www.khanacademy.org/math/algebra/x2f4f7d0/x2f4f7d0/quadratic-functions&gt;

### Future Work

* Investigate the properties of quadratic functions in more detail, such as their domain and range.
* Explore the relationship between quadratic functions and their graphical representation.
* Use quadratic functions to model real-world phenomena and make predictions about the behavior of the system.

### Code

```python
import sympy as sp

# Define the variable
x = sp.symbols(&#39;x&#39;)

# Define the function
f = x**2 - 6*x + 5

# Solve for x
solutions = sp.solve(f, x)

print(solutions)

# Substitute x = 0 into the function
y_intercept = f.subs(x, 0)

print(y_intercept)

# Calculate the x-coordinate of the vertex
vertex_x = -(-6) / (2 * 1)

print(vertex_x)

# Calculate the y-coordinate of the vertex
vertex_y = f.subs(x, vertex_x)

print(vertex_y)

# Calculate the equation of the axis of symmetry
axis_of_symmetry = x - vertex_x

print(axis_of_symmetry)
</code></pre>


                
                    
                        

                    
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