Given The Quadratic Function $y = 2x^2 + 4x - 2$, Determine The Following:1. Calculate $-\frac{b}{2a}$.2. Find The Axis Of Symmetry.3. Determine The Vertex Of The Parabola.

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various problems in algebra, geometry, and other branches of mathematics. In this article, we will analyze the quadratic function $y = 2x^2 + 4x - 2$ and determine its axis of symmetry, vertex, and other key properties.

Calculating $-\frac{b}{2a}$

To calculate $-\frac{b}{2a}$, we need to identify the values of $a$ and $b$ in the given quadratic function. In the function $y = 2x^2 + 4x - 2$, $a = 2$ and $b = 4$. Now, we can substitute these values into the formula:

$$-\frac{b}{2a} = -\frac{4}{2(2)} = -\frac{4}{4} = -1$$

Finding the Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex of the parabola. To find the axis of symmetry, we can use the formula:

$$x = -\frac{b}{2a}$$

We have already calculated the value of $-\frac{b}{2a}$ in the previous section, which is $-1$. Therefore, the axis of symmetry is:

$$x = -\frac{4}{2(2)} = -1$$

Determining the Vertex of the Parabola

The vertex of the parabola is the point where the parabola changes direction. To find the vertex, we can use the formula:

$$x = -\frac{b}{2a}$$

We have already calculated the value of $-\frac{b}{2a}$, which is $-1$. Now, we can substitute this value into the function to find the y-coordinate of the vertex:

$$y = 2(-1)^2 + 4(-1) - 2 = 2 - 4 - 2 = -4$$

Therefore, the vertex of the parabola is:

$$(x, y) = (-1, -4)$$

Graphing the Parabola

To graph the parabola, we can use the axis of symmetry and the vertex. The axis of symmetry is a vertical line that passes through the vertex, and the parabola is symmetric about this line. We can plot the vertex and then use the axis of symmetry to plot the rest of the parabola.

Conclusion

In this article, we analyzed the quadratic function $y = 2x^2 + 4x - 2$ and determined its axis of symmetry, vertex, and other key properties. We calculated $-\frac{b}{2a}$, found the axis of symmetry, and determined the vertex of the parabola. We also graphed the parabola using the axis of symmetry and the vertex.

Key Takeaways

• The axis of symmetry is a vertical line that passes through the vertex of the parabola.
• The vertex of the parabola is the point where the parabola changes direction.
• The quadratic function $y = 2x^2 + 4x - 2$ has an axis of symmetry at $x = -1$ and a vertex at $(-1, -4)$.

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] Larson, R. (2013). College Algebra. Cengage Learning.
• [2] Sullivan, M. (2013). College Algebra. Pearson Education.
• [3] Anton, H. (2013). College Algebra. John Wiley & Sons.
  

Introduction

In our previous article, we analyzed the quadratic function $y = 2x^2 + 4x - 2$ and determined its axis of symmetry, vertex, and other key properties. In this article, we will answer some frequently asked questions about quadratic functions and provide additional insights into their properties.

Q&A

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

A: The axis of symmetry is a vertical line that passes through the vertex of the parabola. It is a key property of quadratic functions and can be used to graph the parabola.

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

A: To find the axis of symmetry, you can use the formula:

$$x = -\frac{b}{2a}$$

where $a$ and $b$ are the coefficients of the quadratic function.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the point where the parabola changes direction. It is the minimum or maximum point of the parabola, depending on the direction of the parabola.

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

A: To find the vertex, you can use the formula:

$$x = -\frac{b}{2a}$$

where $a$ and $b$ are the coefficients of the quadratic function. Then, substitute this value into the function to find the y-coordinate of the vertex.

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

A: A quadratic function is a polynomial function of degree 2, while a linear function is a polynomial function of degree 1. Quadratic functions have a parabolic shape, while linear functions have a straight line shape.

Q: Can a quadratic function have more than one axis of symmetry?

A: No, a quadratic function can only have one axis of symmetry.

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

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

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use the axis of symmetry and the vertex. Plot the vertex and then use the axis of symmetry to plot the rest of the parabola.

Q: What is the significance of the axis of symmetry in quadratic functions?

A: The axis of symmetry is a key property of quadratic functions and can be used to graph the parabola. It is also used to find the vertex and other key properties of the parabola.

Conclusion

In this article, we answered some frequently asked questions about quadratic functions and provided additional insights into their properties. We hope that this article has been helpful in understanding the key concepts of quadratic functions.

Key Takeaways

• The axis of symmetry is a vertical line that passes through the vertex of the parabola.
• The vertex of the parabola is the point where the parabola changes direction.
• Quadratic functions have a parabolic shape and can be graphed using the axis of symmetry and the vertex.
• The axis of symmetry is a key property of quadratic functions and can be used to find the vertex and other key properties of the parabola.

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] Larson, R. (2013). College Algebra. Cengage Learning.
• [2] Sullivan, M. (2013). College Algebra. Pearson Education.
• [3] Anton, H. (2013). College Algebra. John Wiley & Sons.