Which Quadratic Function In Standard Form Has The Values $a = -3.5$, $b = 2.7$, And $c = -8.2$?A. $f(x) = 2.7x^2 - 8.2x - 3.5$B. $f(x) = 2.7x^2 - 3.5x - 8.2$C. $f(x) = -3.5x^2 - 8.2x + 2.7$D.

Which Quadratic Function in Standard Form Has the Values $a = -3.5$, $b = 2.7$, and $c = -8.2$?

Quadratic functions in standard form are a fundamental concept in algebra and mathematics. They are used to model various real-world phenomena, such as the trajectory of a projectile, the growth of a population, and the motion of an object under the influence of a force. A quadratic function in standard form is represented by the equation $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

The Importance of Quadratic Functions in Standard Form

Quadratic functions in standard form are essential in various fields, including physics, engineering, economics, and computer science. They are used to solve problems involving optimization, maximization, and minimization. For instance, in physics, quadratic functions are used to model the motion of an object under the influence of gravity, while in economics, they are used to model the demand and supply of a product.

The Given Values: $a = -3.5$, $b = 2.7$, and $c = -8.2$

We are given the values of $a$, $b$, and $c$ as $-3.5$, $2.7$, and $-8.2$, respectively. We need to determine which quadratic function in standard form corresponds to these values.

Analyzing the Options

Let's analyze the options given:

A. $f(x) = 2.7x^2 - 8.2x - 3.5$ B. $f(x) = 2.7x^2 - 3.5x - 8.2$ C. $f(x) = -3.5x^2 - 8.2x + 2.7$ D. (Not given)

Option A: $f(x) = 2.7x^2 - 8.2x - 3.5$

In option A, the coefficient of $x^2$ is $2.7$, which is equal to $b$. However, the coefficient of $x$ is $-8.2$, which is equal to $c$. The constant term is $-3.5$, which is equal to $a$. Therefore, option A is a possible solution.

Option B: $f(x) = 2.7x^2 - 3.5x - 8.2$

In option B, the coefficient of $x^2$ is $2.7$, which is equal to $b$. However, the coefficient of $x$ is $-3.5$, which is not equal to $c$. The constant term is $-8.2$, which is equal to $c$. Therefore, option B is not a possible solution.

Option C: $f(x) = -3.5x^2 - 8.2x + 2.7$

In option C, the coefficient of $x^2$ is $-3.5$, which is equal to $a$. However, the coefficient of $x$ is $-8.2$, which is equal to $c$. The constant term is $2.7$, which is not equal to $b$. Therefore, option C is not a possible solution.

Conclusion

Based on the analysis of the options, we can conclude that option A is the correct solution. The quadratic function in standard form that corresponds to the values $a = -3.5$, $b = 2.7$, and $c = -8.2$ is $f(x) = 2.7x^2 - 8.2x - 3.5$.

Final Answer

Understanding Quadratic Functions in Standard Form

Quadratic functions in standard form are a fundamental concept in algebra and mathematics. They are used to model various real-world phenomena, such as the trajectory of a projectile, the growth of a population, and the motion of an object under the influence of a force. A quadratic function in standard form is represented by the equation $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Frequently Asked Questions

Q: What is a quadratic function in standard form?

A: A quadratic function in standard form is represented by the equation $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is the significance of quadratic functions in standard form?

A: Quadratic functions in standard form are essential in various fields, including physics, engineering, economics, and computer science. They are used to solve problems involving optimization, maximization, and minimization.

Q: How do I determine the values of $a$, $b$, and $c$ in a quadratic function in standard form?

A: The values of $a$, $b$, and $c$ can be determined by analyzing the graph of the quadratic function or by using algebraic methods.

Q: What is the difference between a quadratic function in standard form and a quadratic function in vertex form?

A: A quadratic function in standard form is represented by the equation $f(x) = ax^2 + bx + c$, while a quadratic function in vertex form is represented by the equation $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: How do I graph a quadratic function in standard form?

A: To graph a quadratic function in standard form, you can use the following steps:

1. Determine the values of $a$, $b$, and $c$.
2. Find the x-intercepts of the parabola by setting $f(x) = 0$ and solving for $x$.
3. Find the y-intercept of the parabola by evaluating $f(0)$.
4. Plot the x-intercepts and the y-intercept on a coordinate plane.
5. Draw a smooth curve through the points to form the graph of the parabola.

Q: How do I solve a quadratic equation in standard form?

A: To solve a quadratic equation in standard form, you can use the following steps:

1. Determine the values of $a$, $b$, and $c$.
2. Use the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
3. Simplify the expression to find the solutions.

Q: What is the difference between a quadratic function in standard form and a quadratic function in factored form?

A: A quadratic function in standard form is represented by the equation $f(x) = ax^2 + bx + c$, while a quadratic function in factored form is represented by the equation $f(x) = a(x - r_1)(x - r_2)$, where $r_1$ and $r_2$ are the roots of the quadratic equation.

Q: How do I convert a quadratic function in standard form to a quadratic function in factored form?

A: To convert a quadratic function in standard form to a quadratic function in factored form, you can use the following steps:

1. Determine the values of $a$, $b$, and $c$.
2. Factor the quadratic expression: $f(x) = a(x - r_1)(x - r_2)$.
3. Simplify the expression to find the factored form.

Conclusion

Quadratic functions in standard form are a fundamental concept in algebra and mathematics. They are used to model various real-world phenomena and are essential in various fields, including physics, engineering, economics, and computer science. By understanding the properties and applications of quadratic functions in standard form, you can solve problems involving optimization, maximization, and minimization.

Final Answer

The final answer is that quadratic functions in standard form are a powerful tool for modeling and solving problems in various fields. By understanding the properties and applications of quadratic functions in standard form, you can solve problems involving optimization, maximization, and minimization.