Given $f(x) = -x^2 + 2x$, Find $f(-3$\].

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Introduction

Quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields, including physics, engineering, and economics. In this article, we will focus on finding the value of a quadratic function at a given point. We will use the function f(x)=βˆ’x2+2xf(x) = -x^2 + 2x as an example and find its value at x=βˆ’3x = -3.

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means that the highest power of the variable is two. The general form of a quadratic function is f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants. The graph of a quadratic function is a parabola, which is a U-shaped curve.

The Function f(x)=βˆ’x2+2xf(x) = -x^2 + 2x

The function f(x)=βˆ’x2+2xf(x) = -x^2 + 2x is a quadratic function with a negative leading coefficient. This means that the parabola opens downward, and the vertex of the parabola is the maximum point.

Finding the Value of f(βˆ’3)f(-3)

To find the value of f(βˆ’3)f(-3), we need to substitute x=βˆ’3x = -3 into the function f(x)=βˆ’x2+2xf(x) = -x^2 + 2x. This means that we need to replace xx with βˆ’3-3 in the function and simplify the expression.

Step 1: Substitute x=βˆ’3x = -3 into the Function

f(βˆ’3)=βˆ’(βˆ’3)2+2(βˆ’3)f(-3) = -( -3 )^2 + 2( -3 )

Step 2: Simplify the Expression

f(βˆ’3)=βˆ’(βˆ’3)2+2(βˆ’3)f(-3) = -(-3)^2 + 2(-3)

f(βˆ’3)=βˆ’9βˆ’6f(-3) = -9 - 6

f(βˆ’3)=βˆ’15f(-3) = -15

Conclusion

In this article, we found the value of the quadratic function f(x)=βˆ’x2+2xf(x) = -x^2 + 2x at x=βˆ’3x = -3. We used the function and substituted x=βˆ’3x = -3 into it, and then simplified the expression to find the value of f(βˆ’3)f(-3). The value of f(βˆ’3)f(-3) is βˆ’15-15.

Applications of Quadratic Functions

Quadratic functions have numerous applications in various fields, including physics, engineering, and economics. Some of the applications of quadratic functions include:

  • Projectile Motion: Quadratic functions are used to model the trajectory of projectiles, such as the path of a thrown ball or the trajectory of a rocket.
  • Optimization: Quadratic functions are used to optimize functions, such as finding the maximum or minimum value of a function.
  • Economics: Quadratic functions are used to model the behavior of economic systems, such as the demand and supply curves of a product.

Examples of Quadratic Functions

Some examples of quadratic functions include:

  • Parabolas: The graph of a quadratic function is a parabola, which is a U-shaped curve.
  • Circles: The equation of a circle is a quadratic function, which is used to model the shape of a circle.
  • Ellipses: The equation of an ellipse is a quadratic function, which is used to model the shape of an ellipse.

Tips for Working with Quadratic Functions

When working with quadratic functions, it is essential to remember the following tips:

  • Use the correct notation: When working with quadratic functions, it is essential to use the correct notation, which includes the use of parentheses and the correct order of operations.
  • Simplify expressions: When working with quadratic functions, it is essential to simplify expressions to find the value of the function.
  • Use algebraic techniques: When working with quadratic functions, it is essential to use algebraic techniques, such as factoring and the quadratic formula, to solve equations and find the value of the function.

Conclusion

In conclusion, quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we found the value of the quadratic function f(x)=βˆ’x2+2xf(x) = -x^2 + 2x at x=βˆ’3x = -3. We used the function and substituted x=βˆ’3x = -3 into it, and then simplified the expression to find the value of f(βˆ’3)f(-3). The value of f(βˆ’3)f(-3) is βˆ’15-15. We also discussed the applications of quadratic functions, examples of quadratic functions, and tips for working with quadratic functions.

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Frequently Asked Questions

Q: What is a quadratic function?

A: A quadratic function is a polynomial function of degree two, which means that the highest power of the variable is two. The general form of a quadratic function is f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants.

Q: What is the graph of a quadratic function?

A: The graph of a quadratic function is a parabola, which is a U-shaped curve. The parabola can open upward or downward, depending on the sign of the leading coefficient.

Q: How do I find the value of a quadratic function at a given point?

A: To find the value of a quadratic function at a given point, you need to substitute the value of the point into the function and simplify the expression. For example, to find the value of f(x)=βˆ’x2+2xf(x) = -x^2 + 2x at x=βˆ’3x = -3, you would substitute x=βˆ’3x = -3 into the function and simplify the expression.

Q: What are some common applications of quadratic functions?

A: Quadratic functions have numerous applications in various fields, including physics, engineering, and economics. Some of the applications of quadratic functions include:

  • Projectile Motion: Quadratic functions are used to model the trajectory of projectiles, such as the path of a thrown ball or the trajectory of a rocket.
  • Optimization: Quadratic functions are used to optimize functions, such as finding the maximum or minimum value of a function.
  • Economics: Quadratic functions are used to model the behavior of economic systems, such as the demand and supply curves of a product.

Q: How do I factor a quadratic function?

A: To factor a quadratic function, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term. For example, to factor the quadratic function f(x)=x2+5x+6f(x) = x^2 + 5x + 6, you would find the two numbers 22 and 33 whose product is 66 and whose sum is 55.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. The quadratic formula is given by:

x=βˆ’bΒ±b2βˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the coefficients of the quadratic function.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to substitute the values of aa, bb, and cc into the formula and simplify the expression. For example, to solve the quadratic equation x2+5x+6=0x^2 + 5x + 6 = 0, you would substitute a=1a = 1, b=5b = 5, and c=6c = 6 into the quadratic formula and simplify the expression.

Q: What are some common mistakes to avoid when working with quadratic functions?

A: Some common mistakes to avoid when working with quadratic functions include:

  • Not using the correct notation: When working with quadratic functions, it is essential to use the correct notation, which includes the use of parentheses and the correct order of operations.
  • Not simplifying expressions: When working with quadratic functions, it is essential to simplify expressions to find the value of the function.
  • Not using algebraic techniques: When working with quadratic functions, it is essential to use algebraic techniques, such as factoring and the quadratic formula, to solve equations and find the value of the function.

Conclusion

In conclusion, quadratic functions are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we answered some frequently asked questions about quadratic functions, including what a quadratic function is, how to find the value of a quadratic function at a given point, and how to factor a quadratic function. We also discussed the quadratic formula and some common mistakes to avoid when working with quadratic functions.