Overview of Quadratic Functions

Oct 13, 2025 by windailys.com 33 views

Overview of Quadratic Functions

Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. 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.

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. In this form, the vertex is the minimum or maximum point of the parabola, depending on the sign of $a$ . If $a$ is positive, the vertex is the minimum point, and if $a$ is negative, the vertex is the maximum point.

Given Information

We are given that the vertex of the parabola is at $(2, -25)$ and one of the $x$ -intercepts is at $(7, 0)$ . Since the vertex is at $(2, -25)$ , we know that the equation of the quadratic function can be written in the form $f(x) = a(x - 2)^2 - 25$ . We also know that one of the $x$ -intercepts is at $(7, 0)$ , which means that when $x = 7$ , $f(x) = 0$ .

Finding the Value of $a$

To find the value of $a$ , we can substitute $x = 7$ into the equation $f(x) = a(x - 2)^2 - 25$ and solve for $a$ . This gives us:

0 = a(7 - 2)^2 - 25

Simplifying the equation, we get:

0 = a(5)^2 - 25

0 = 25a - 25

25 = 25a

a = 1

Writing the Equation of the Quadratic Function

Now that we have found the value of $a$ , we can write the equation of the quadratic function. Substituting $a = 1$ into the equation $f(x) = a(x - 2)^2 - 25$ , we get:

f(x) = (x - 2)^2 - 25

Expanding the squared term, we get:

f(x) = x^2 - 4x + 4 - 25

Simplifying the equation, we get:

f(x) = x^2 - 4x - 21

Factoring the Quadratic Function

We can factor the quadratic function $f(x) = x^2 - 4x - 21$ by finding two numbers whose product is $-21$ and whose sum is $-4$ . The numbers are $-7$ and $3$ , so we can write the quadratic function as:

f(x) = (x - 7)(x + 3)

However, this is not the only possible factorization. We can also write the quadratic function as:

f(x) = (x - 3)(x + 7)

Conclusion

In conclusion, the equation of the quadratic function with a vertex at $(2, -25)$ and an $x$ -intercept at $(7, 0)$ is $f(x) = (x - 3)(x + 7)$ . This is one of the possible factorizations of the quadratic function, and it satisfies the given conditions.

Final Answer

The final answer is: $\boxed{f(x) = (x - 3)(x + 7)}$

Overview of Quadratic Functions

Frequently Asked Questions

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. In this form, the vertex is the minimum or maximum point of the parabola, depending on the sign of $a$ . If $a$ is positive, the vertex is the minimum point, and if $a$ is negative, the vertex is the maximum point.

Q: How do I find the equation of a quadratic function with a given vertex and x-intercept?

A: To find the equation of a quadratic function with a given vertex and x-intercept, you can use the vertex form of a quadratic function. The vertex form is given by $f(x) = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola. You can substitute the given vertex into this equation and solve for $a$ . Then, you can use the x-intercept to find the value of $a$ .

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

A: The x-intercept of a quadratic function is the point where the graph of the function intersects the x-axis. This point is also known as the root or solution of the equation. The x-intercept is an important concept in quadratic functions, as it can be used to find the equation of the function.

Q: How do I factor a quadratic function?

A: To factor a quadratic function, you can use the factoring method, which involves finding two numbers whose product is the constant term and whose sum is the coefficient of the linear term. You can also use the quadratic formula to factor the function.

Q: What is the difference between the vertex form and the standard 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. The standard form of a quadratic function is given by $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. The vertex form is more convenient for graphing and analyzing the function, while the standard form is more convenient for solving equations.

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

A: Yes, a quadratic function can have more than one x-intercept. In fact, a quadratic function can have up to two x-intercepts, depending on the values of the coefficients.

Q: How do I determine the number of x-intercepts of a quadratic function?

A: To determine the number of x-intercepts of a quadratic function, you can use the discriminant, which is given by $b^2 - 4ac$ . If the discriminant is positive, the function has two x-intercepts. If the discriminant is zero, the function has one x-intercept. If the discriminant is negative, the function has no x-intercepts.

Conclusion

In conclusion, quadratic functions are an important concept in mathematics, and they have numerous applications in various fields. Understanding the equation of a quadratic function with a vertex and an x-intercept requires knowledge of the vertex form and the standard form of a quadratic function. By using the vertex form and the standard form, you can find the equation of a quadratic function and analyze its properties.

Final Answer

The final answer is: $\boxed{f(x) = (x - 3)(x + 7)}$