What Are The $x$-intercept And Vertex Of This Quadratic Function?$g(x) = -5(x-3)^2$Write Each Feature As An Ordered Pair: $(a, B)$.The $x$-intercept Of Function $g$ Is $\square$.The

Mar 11, 2025 by ADMIN 194 views

What are the $x$ -intercept and vertex of this quadratic function?

Understanding Quadratic Functions

Quadratic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $a$ cannot be zero.

The Given Quadratic Function

In this article, we are given a quadratic function $g(x) = -5(x-3)^2$ . Our goal is to find the $x$ -intercept and vertex of this function. To do this, we need to understand the properties of quadratic functions and how to find their key features.

Finding the $x$ -intercept

The $x$ -intercept of a function is the point where the function intersects the $x$ -axis. In other words, it is the value of $x$ when the function is equal to zero. To find the $x$ -intercept of the given function, we need to set $g(x)$ equal to zero and solve for $x$ .

import sympy as sp
x = sp.symbols('x')

g = -5*(x-3)**2

x_intercept = sp.solve(g, x)
print(x_intercept)

Finding the Vertex

The vertex of a quadratic function is the point where the function reaches its maximum or minimum value. The vertex form of a quadratic function is $f(x) = a(x-h)^2 + k$ , where $(h, k)$ is the vertex. To find the vertex of the given function, we need to rewrite it in vertex form.

import sympy as sp

x = sp.symbols('x')

g = -5*(x-3)**2

vertex_form = sp.expand(g)
print(vertex_form)

The $x$ -intercept and Vertex

From the previous sections, we have found that the $x$ -intercept of the function is $x = 3$ , and the vertex is $(3, -45)$ .

Conclusion

In this article, we have discussed the $x$ -intercept and vertex of a quadratic function. We have used Python code to find the $x$ -intercept and vertex of the given function. The $x$ -intercept is the point where the function intersects the $x$ -axis, and the vertex is the point where the function reaches its maximum or minimum value. Understanding these key features of quadratic functions is essential in various fields, including algebra, geometry, and calculus.

The $x$ -intercept of function $g$ is $(3, 0)$

The vertex of function $g$ is $(3, -45)$

Key Takeaways

The $x$ -intercept of a quadratic function is the point where the function intersects the $x$ -axis.
The vertex of a quadratic function is the point where the function reaches its maximum or minimum value.
Understanding the $x$ -intercept and vertex of a quadratic function is essential in various fields, including algebra, geometry, and calculus.

Further Reading

For further reading on quadratic functions, we recommend the following resources:

Khan Academy: Quadratic Functions
Mathway: Quadratic Functions
Wolfram MathWorld: Quadratic Functions

References

[1] Khan Academy. Quadratic Functions. Retrieved from https://www.khanacademy.org/math/algebra/quadratic-equations
[2] Mathway. Quadratic Functions. Retrieved from https://www.mathway.com/subjects/quadratic-functions
[3] Wolfram MathWorld. Quadratic Functions. Retrieved from https://mathworld.wolfram.com/QuadraticFunction.html
Quadratic Function Q&A

Understanding Quadratic Functions

Frequently Asked Questions

Q: What is the $x$ -intercept of a quadratic function?

A: The $x$ -intercept of a quadratic function is the point where the function intersects the $x$ -axis. In other words, it is the value of $x$ when the function is equal to zero.

Q: How do I find the $x$ -intercept of a quadratic function?

A: To find the $x$ -intercept of a quadratic function, you need to set the function equal to zero and solve for $x$ . You can use algebraic methods or graphing tools to find the $x$ -intercept.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point where the function reaches its maximum or minimum value. The vertex form of a quadratic function is $f(x) = a(x-h)^2 + k$ , where $(h, k)$ is the vertex.

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

A: To find the vertex of a quadratic function, you need to rewrite the function in vertex form. You can use algebraic methods or graphing tools to find the vertex.

Q: What is the difference between the $x$ -intercept and the vertex of a quadratic function?

A: The $x$ -intercept of a quadratic function is the point where the function intersects the $x$ -axis, while the vertex is the point where the function reaches its maximum or minimum value.

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

A: No, a quadratic function can have at most two $x$ -intercepts.

Q: Can a quadratic function have more than one vertex?

A: No, a quadratic function can have at most one vertex.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use algebraic methods or graphing tools. You can also use the $x$ -intercept and vertex to graph the function.

Q: What is the significance of the $x$ -intercept and vertex of a quadratic function?

A: The $x$ -intercept and vertex of a quadratic function are important because they help us understand the behavior of the function. The $x$ -intercept tells us where the function intersects the $x$ -axis, while the vertex tells us where the function reaches its maximum or minimum value.

Q: Can a quadratic function be used to model real-world phenomena?

A: Yes, quadratic functions can be used to model real-world phenomena, such as the trajectory of a projectile or the growth of a population.