Find The Vertex Of The Quadratic Function $f(x) = 2x^2 - 6x + 7$.A. \[$(0, 7)\$\]B. \[$(1, 2)\$\]C. \[$(-1.5, 2.5)\$\]D. \[$(1.5, 2.5)\$\]

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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 f(x)=ax2+bx+cf(x) = ax^2 + bx + c, where aa, bb, and cc are constants. One of the key concepts in quadratic functions is the vertex, which is the maximum or minimum point of the parabola represented by the function. In this article, we will focus on finding the vertex of the quadratic function f(x)=2x2−6x+7f(x) = 2x^2 - 6x + 7.

Understanding the Vertex Form

The vertex form of a quadratic function is given by f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola. To find the vertex, we need to convert the given function into vertex form. The process involves completing the square, which is a technique used to rewrite a quadratic expression in a perfect square form.

Completing the Square

To complete the square, we start by factoring out the coefficient of the x2x^2 term, which is 2 in this case. We then take half of the coefficient of the xx term, square it, and add it to both sides of the equation.

f(x) = 2x^2 - 6x + 7
f(x) = 2(x^2 - 3x) + 7

Finding the Value of h

To find the value of hh, we need to take half of the coefficient of the xx term and square it. In this case, the coefficient of the xx term is -3. Half of -3 is -1.5, and squaring it gives us 2.25.

h = (-3/2)^2 = 2.25

Finding the Value of k

Now that we have the value of hh, we can find the value of kk by substituting hh into the equation. We get:

f(x) = 2(x - 1.5)^2 + 7 - 2(1.5)^2
f(x) = 2(x - 1.5)^2 + 7 - 4.5
f(x) = 2(x - 1.5)^2 + 2.5

Finding the Vertex

Now that we have the vertex form of the function, we can find the vertex by identifying the values of hh and kk. In this case, the vertex is (h,k)=(−1.5,2.5)(h, k) = (-1.5, 2.5).

Conclusion

In this article, we have learned how to find the vertex of a quadratic function using the vertex form. We have also seen how to complete the square to convert the given function into vertex form. By following these steps, we can find the vertex of any quadratic function.

Answer

The correct answer is C. {(-1.5, 2.5)$}$

Discussion

The vertex of a quadratic function is an important concept in mathematics, and it has many real-world applications. For example, in physics, the vertex of a quadratic function can represent the maximum or minimum point of a projectile's trajectory. In economics, the vertex of a quadratic function can represent the maximum or minimum point of a company's profit or loss.

Related Topics

  • Quadratic Functions: Quadratic functions are polynomial functions of degree two, which means the highest power of the variable is two.
  • Vertex Form: The vertex form of a quadratic function is given by f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.
  • Completing the Square: Completing the square is a technique used to rewrite a quadratic expression in a perfect square form.
  • Graphing Quadratic Functions: Graphing quadratic functions involves plotting the points on a coordinate plane and drawing a smooth curve through them.

Practice Problems

  • Find the vertex of the quadratic function f(x)=x2+4x+3f(x) = x^2 + 4x + 3.
  • Find the vertex of the quadratic function f(x)=2x2−5x−3f(x) = 2x^2 - 5x - 3.
  • Find the vertex of the quadratic function f(x)=x2−2x−1f(x) = x^2 - 2x - 1.

Solutions

  • The vertex of the quadratic function f(x)=x2+4x+3f(x) = x^2 + 4x + 3 is (h,k)=(−2,1)(h, k) = (-2, 1).
  • The vertex of the quadratic function f(x)=2x2−5x−3f(x) = 2x^2 - 5x - 3 is (h,k)=(1.25,−4.75)(h, k) = (1.25, -4.75).
  • The vertex of the quadratic function f(x)=x2−2x−1f(x) = x^2 - 2x - 1 is (h,k)=(1,−2)(h, k) = (1, -2).

Conclusion

In conclusion, finding the vertex of a quadratic function is an important concept in mathematics, and it has many real-world applications. By following the steps outlined in this article, we can find the vertex of any quadratic function.

Introduction

In our previous article, we discussed how to find the vertex of a quadratic function using the vertex form. In this article, we will answer some of the most frequently asked questions about quadratic function vertices.

Q: What is the vertex of a quadratic function?

A: The vertex of a quadratic function is the maximum or minimum point of the parabola represented by the function.

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

A: To find the vertex of a quadratic function, you need to convert the function into vertex form using the process of completing the square.

Q: What is completing the square?

A: Completing the square is a technique used to rewrite a quadratic expression in a perfect square form. It involves factoring out the coefficient of the x2x^2 term, taking half of the coefficient of the xx term, squaring it, and adding it to both sides of the equation.

Q: How do I find the value of h in the vertex form?

A: To find the value of hh, you need to take half of the coefficient of the xx term and square it.

Q: How do I find the value of k in the vertex form?

A: To find the value of kk, you need to substitute the value of hh into the equation and simplify.

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

A: The vertex of a quadratic function is significant because it represents the maximum or minimum point of the parabola. It has many real-world applications, such as in physics, economics, and engineering.

Q: Can I use the vertex form to graph a quadratic function?

A: Yes, you can use the vertex form to graph a quadratic function. By plotting the points on a coordinate plane and drawing a smooth curve through them, you can visualize the parabola represented by the function.

Q: How do I find the x-intercepts of a quadratic function?

A: To find the x-intercepts of a quadratic function, you need to set the function equal to zero and solve for xx.

Q: How do I find the y-intercept of a quadratic function?

A: To find the y-intercept of a quadratic function, you need to substitute x=0x = 0 into the function and simplify.

Q: Can I use the vertex form to find the equation of a quadratic function?

A: Yes, you can use the vertex form to find the equation of a quadratic function. By substituting the values of hh and kk into the vertex form, you can obtain the equation of the parabola.

Q: What are some real-world applications of quadratic function vertices?

A: Quadratic function vertices have many real-world applications, such as in physics, economics, and engineering. For example, in physics, the vertex of a quadratic function can represent the maximum or minimum point of a projectile's trajectory. In economics, the vertex of a quadratic function can represent the maximum or minimum point of a company's profit or loss.

Conclusion

In conclusion, quadratic function vertices are an important concept in mathematics, and they have many real-world applications. By understanding how to find the vertex of a quadratic function, you can visualize the parabola represented by the function and solve problems in various fields.

Practice Problems

  • Find the vertex of the quadratic function f(x)=x2+4x+3f(x) = x^2 + 4x + 3.
  • Find the vertex of the quadratic function f(x)=2x2−5x−3f(x) = 2x^2 - 5x - 3.
  • Find the vertex of the quadratic function f(x)=x2−2x−1f(x) = x^2 - 2x - 1.

Solutions

  • The vertex of the quadratic function f(x)=x2+4x+3f(x) = x^2 + 4x + 3 is (h,k)=(−2,1)(h, k) = (-2, 1).
  • The vertex of the quadratic function f(x)=2x2−5x−3f(x) = 2x^2 - 5x - 3 is (h,k)=(1.25,−4.75)(h, k) = (1.25, -4.75).
  • The vertex of the quadratic function f(x)=x2−2x−1f(x) = x^2 - 2x - 1 is (h,k)=(1,−2)(h, k) = (1, -2).

Discussion

Quadratic function vertices are an important concept in mathematics, and they have many real-world applications. By understanding how to find the vertex of a quadratic function, you can visualize the parabola represented by the function and solve problems in various fields.

Related Topics

  • Quadratic Functions: Quadratic functions are polynomial functions of degree two, which means the highest power of the variable is two.
  • Vertex Form: The vertex form of a quadratic function is given by f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.
  • Completing the Square: Completing the square is a technique used to rewrite a quadratic expression in a perfect square form.
  • Graphing Quadratic Functions: Graphing quadratic functions involves plotting the points on a coordinate plane and drawing a smooth curve through them.

Glossary

  • Vertex: The vertex of a quadratic function is the maximum or minimum point of the parabola represented by the function.
  • Completing the Square: Completing the square is a technique used to rewrite a quadratic expression in a perfect square form.
  • Vertex Form: The vertex form of a quadratic function is given by f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k, where (h,k)(h, k) is the vertex of the parabola.
  • Quadratic Function: A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two.