The Quadratic Function Y = 0.4 ( X − 0.5 ) 2 − 2 Y = 0.4(x - 0.5)^2 - 2 Y = 0.4 ( X − 0.5 ) 2 − 2 Has A Vertex At The Point ( 0.5 , − 2 (0.5, -2 ( 0.5 , − 2 ] And Opens Upward.

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Introduction

In mathematics, 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 y=ax2+bx+cy = ax^2 + bx + c, where aa, bb, and cc are constants. In this article, we will focus on the quadratic function y=0.4(x−0.5)2−2y = 0.4(x - 0.5)^2 - 2 and explore its properties and behavior.

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola. In the given quadratic function y=0.4(x−0.5)2−2y = 0.4(x - 0.5)^2 - 2, we can see that the vertex is at the point (0.5,−2)(0.5, -2). This means that the vertex form of the quadratic function is y=0.4(x−0.5)2−2y = 0.4(x - 0.5)^2 - 2.

Properties of the Quadratic Function

The quadratic function y=0.4(x−0.5)2−2y = 0.4(x - 0.5)^2 - 2 has several properties that can be determined from its vertex form. Some of these properties include:

  • Vertex: The vertex of the parabola is at the point (0.5,−2)(0.5, -2).
  • Axis of Symmetry: The axis of symmetry of the parabola is the vertical line x=0.5x = 0.5.
  • Direction of Opening: The parabola opens upward, which means that the coefficient of the squared term aa is positive.
  • Minimum or Maximum Value: Since the parabola opens upward, the vertex represents the minimum value of the function.

Graph of the Quadratic Function

The graph of the quadratic function y=0.4(x−0.5)2−2y = 0.4(x - 0.5)^2 - 2 is a parabola that opens upward. The vertex of the parabola is at the point (0.5,−2)(0.5, -2), and the axis of symmetry is the vertical line x=0.5x = 0.5. The graph of the quadratic function can be sketched using the following steps:

  1. Plot the Vertex: Plot the vertex of the parabola at the point (0.5,−2)(0.5, -2).
  2. Plot the Axis of Symmetry: Plot the axis of symmetry as the vertical line x=0.5x = 0.5.
  3. Plot the Parabola: Plot the parabola by connecting the points on either side of the axis of symmetry.

Solving Quadratic Equations

Quadratic equations are equations of the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants. The quadratic function y=0.4(x−0.5)2−2y = 0.4(x - 0.5)^2 - 2 can be used to solve quadratic equations of the form 0.4(x−0.5)2−2=00.4(x - 0.5)^2 - 2 = 0. To solve this equation, we can set the quadratic function equal to zero and solve for xx.

Solving the Quadratic Equation

To solve the quadratic equation 0.4(x−0.5)2−2=00.4(x - 0.5)^2 - 2 = 0, we can start by adding 22 to both sides of the equation:

0.4(x−0.5)2=20.4(x - 0.5)^2 = 2

Next, we can divide both sides of the equation by 0.40.4:

(x−0.5)2=5(x - 0.5)^2 = 5

Now, we can take the square root of both sides of the equation:

x−0.5=±5x - 0.5 = \pm \sqrt{5}

Finally, we can add 0.50.5 to both sides of the equation to solve for xx:

x=0.5±5x = 0.5 \pm \sqrt{5}

Conclusion

In this article, we have explored the properties and behavior of the quadratic function y=0.4(x−0.5)2−2y = 0.4(x - 0.5)^2 - 2. We have determined that the vertex of the parabola is at the point (0.5,−2)(0.5, -2) and that the parabola opens upward. We have also solved the quadratic equation 0.4(x−0.5)2−2=00.4(x - 0.5)^2 - 2 = 0 using the quadratic function.

References

Frequently Asked Questions About Quadratic Functions

In this article, we will answer some of the most frequently asked questions about quadratic functions.

Q: What is a quadratic function?

A: 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 y=ax2+bx+cy = ax^2 + bx + c, where aa, bb, and cc are constants.

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is given by y=a(x−h)2+ky = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.

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

A: To find the vertex of a quadratic function, you can use the formula h=−b2ah = -\frac{b}{2a} and k=c−b24ak = c - \frac{b^2}{4a}.

Q: What is the axis of symmetry of a quadratic function?

A: The axis of symmetry of a quadratic function is the vertical line that passes through the vertex of the parabola.

Q: How do I determine if a quadratic function opens upward or downward?

A: To determine if a quadratic function opens upward or downward, you can look at the coefficient of the squared term aa. If aa is positive, the parabola opens upward. If aa is negative, the parabola opens downward.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula: x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Q: What is the difference between a quadratic function and a quadratic equation?

A: A quadratic function is a polynomial function of degree two, while a quadratic equation is an equation of the form ax2+bx+c=0ax^2 + bx + c = 0.

Q: Can I use a quadratic function to model real-world situations?

A: Yes, quadratic functions can be used to model real-world situations such as the trajectory of a projectile, the motion of an object under the influence of gravity, and the growth of a population.

Q: How do I graph a quadratic function?

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

  1. Plot the Vertex: Plot the vertex of the parabola at the point (h,k)(h, k).
  2. Plot the Axis of Symmetry: Plot the axis of symmetry as the vertical line x=hx = h.
  3. Plot the Parabola: Plot the parabola by connecting the points on either side of the axis of symmetry.

Q: Can I use a quadratic function to solve optimization problems?

A: Yes, quadratic functions can be used to solve optimization problems such as finding the maximum or minimum value of a function.

Conclusion

In this article, we have answered some of the most frequently asked questions about quadratic functions. We hope that this article has provided you with a better understanding of quadratic functions and how they can be used to model real-world situations.

References