The Axis Of Symmetry For The Function $f(x) = -2x^2 + 4x + 1$ Is The Line $x = 1$. Where Is The Vertex Of The Function Located?A. (0, 1) B. (1, 3) C. (1, 7) D. (2, 1)

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Understanding the Axis of Symmetry

The axis of symmetry is a line that passes through the vertex of a quadratic function and is perpendicular to the x-axis. It is a key concept in algebra and is used to find the vertex of a quadratic function. In this article, we will explore the relationship between the axis of symmetry and the vertex of a quadratic function.

The Given Function

The given function is $f(x) = -2x^2 + 4x + 1$. We are told that the axis of symmetry for this function is the line $x = 1$. This means that the vertex of the function is located on the line $x = 1$.

Finding the Vertex

To find the vertex of the function, we need to use the formula for the x-coordinate of the vertex, which is given by:

x=−b2ax = -\frac{b}{2a}

where $a$ and $b$ are the coefficients of the quadratic function. In this case, $a = -2$ and $b = 4$.

Plugging these values into the formula, we get:

x=−42(−2)x = -\frac{4}{2(-2)}

Simplifying, we get:

x=−4−4x = -\frac{4}{-4}

x=1x = 1

This confirms that the x-coordinate of the vertex is indeed $x = 1$.

Finding the y-coordinate of the Vertex

Now that we have the x-coordinate of the vertex, we can find the y-coordinate by plugging the x-coordinate into the function:

f(x)=−2x2+4x+1f(x) = -2x^2 + 4x + 1

f(1)=−2(1)2+4(1)+1f(1) = -2(1)^2 + 4(1) + 1

f(1)=−2+4+1f(1) = -2 + 4 + 1

f(1)=3f(1) = 3

Therefore, the y-coordinate of the vertex is $y = 3$.

Conclusion

In conclusion, the vertex of the function $f(x) = -2x^2 + 4x + 1$ is located at the point $(1, 3)$. This is confirmed by the fact that the axis of symmetry is the line $x = 1$, and the x-coordinate of the vertex is indeed $x = 1$. The y-coordinate of the vertex is $y = 3$.

Answer

The correct answer is B. (1, 3).

Additional Information

  • The axis of symmetry is a line that passes through the vertex of a quadratic function and is perpendicular to the x-axis.
  • The x-coordinate of the vertex can be found using the formula $x = -\frac{b}{2a}$.
  • The y-coordinate of the vertex can be found by plugging the x-coordinate into the function.

References

  • [1] Algebra, 2nd ed. by Michael Artin
  • [2] Calculus, 3rd ed. by Michael Spivak

Discussion

Q: What is the axis of symmetry, and how is it related to the vertex of a quadratic function?

A: The axis of symmetry is a line that passes through the vertex of a quadratic function and is perpendicular to the x-axis. It is a key concept in algebra and is used to find the vertex of a quadratic function.

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

A: To find the axis of symmetry of a quadratic function, you need to use the formula:

x=−b2ax = -\frac{b}{2a}

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

Q: What is the relationship between the axis of symmetry and the vertex of a quadratic function?

A: The axis of symmetry passes through the vertex of a quadratic function. This means that the vertex is located on the axis of symmetry.

Q: How do you find the x-coordinate of the vertex of a quadratic function?

A: To find the x-coordinate of the vertex of a quadratic function, you need to use the formula:

x=−b2ax = -\frac{b}{2a}

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

Q: How do you find the y-coordinate of the vertex of a quadratic function?

A: To find the y-coordinate of the vertex of a quadratic function, you need to plug the x-coordinate into the function.

Q: What is the significance of the axis of symmetry in algebra?

A: The axis of symmetry is a key concept in algebra and is used to find the vertex of a quadratic function. It is also used to graph quadratic functions and to solve quadratic equations.

Q: Can you give an example of how to find the axis of symmetry and the vertex of a quadratic function?

A: Let's consider the quadratic function:

f(x)=−2x2+4x+1f(x) = -2x^2 + 4x + 1

To find the axis of symmetry, we need to use the formula:

x=−b2ax = -\frac{b}{2a}

where $a = -2$ and $b = 4$.

Plugging these values into the formula, we get:

x=−42(−2)x = -\frac{4}{2(-2)}

Simplifying, we get:

x=−4−4x = -\frac{4}{-4}

x=1x = 1

This confirms that the axis of symmetry is the line $x = 1$.

To find the vertex, we need to plug the x-coordinate into the function:

f(x)=−2x2+4x+1f(x) = -2x^2 + 4x + 1

f(1)=−2(1)2+4(1)+1f(1) = -2(1)^2 + 4(1) + 1

f(1)=−2+4+1f(1) = -2 + 4 + 1

f(1)=3f(1) = 3

Therefore, the vertex is located at the point $(1, 3)$.

Q: What are some common mistakes to avoid when finding the axis of symmetry and the vertex of a quadratic function?

A: Some common mistakes to avoid when finding the axis of symmetry and the vertex of a quadratic function include:

  • Not using the correct formula for the axis of symmetry
  • Not plugging the x-coordinate into the function to find the y-coordinate of the vertex
  • Not simplifying the expression for the axis of symmetry
  • Not checking the work for errors

Q: How can you use the axis of symmetry and the vertex of a quadratic function to solve quadratic equations?

A: The axis of symmetry and the vertex of a quadratic function can be used to solve quadratic equations by:

  • Using the axis of symmetry to find the x-coordinate of the vertex
  • Plugging the x-coordinate into the function to find the y-coordinate of the vertex
  • Using the vertex to find the solutions to the quadratic equation

Q: What are some real-world applications of the axis of symmetry and the vertex of a quadratic function?

A: The axis of symmetry and the vertex of a quadratic function have many real-world applications, including:

  • Modeling population growth and decline
  • Modeling the motion of objects under the influence of gravity
  • Modeling the spread of diseases
  • Modeling the growth of economies

Conclusion

In conclusion, the axis of symmetry and the vertex of a quadratic function are key concepts in algebra that are used to find the vertex of a quadratic function and to solve quadratic equations. By understanding the axis of symmetry and the vertex of a quadratic function, you can solve a wide range of problems in algebra and beyond.