Find The Minimum Value Of The Function, And Then Determine That Value.$\[ Y = 3x^2 - 12x + 1 \\]

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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 given by:

y=ax2+bx+c{ y = ax^2 + bx + c }

where a, b, and c are constants, and x is the variable. In this article, we will focus on finding the minimum value of a quadratic function, and then determine that value.

Understanding Quadratic Functions

Quadratic functions can be represented graphically as a parabola, which is a U-shaped curve. The parabola can open upwards or downwards, depending on the value of the coefficient 'a'. If 'a' is positive, the parabola opens upwards, and if 'a' is negative, the parabola opens downwards.

Graphical Representation

The graph of a quadratic function can be represented as:

y=ax2+bx+c{ y = ax^2 + bx + c }

where a, b, and c are constants. The graph of a quadratic function is a parabola, which can be represented as:

  • A parabola that opens upwards: y = a(x - h)^2 + k
  • A parabola that opens downwards: y = -a(x - h)^2 + k

where (h, k) is the vertex of the parabola.

Finding the Minimum Value

The minimum value of a quadratic function is the value of the function at the vertex of the parabola. To find the minimum value, we need to find the vertex of the parabola.

Vertex Formula

The vertex of a parabola can be found using the formula:

h=−b2a{ h = -\frac{b}{2a} }

where h is the x-coordinate of the vertex, and a and b are the coefficients of the quadratic function.

Finding the Minimum Value

Once we have found the x-coordinate of the vertex, we can find the y-coordinate by substituting the value of x into the quadratic function.

Example

Let's consider the quadratic function:

y=3x2−12x+1{ y = 3x^2 - 12x + 1 }

To find the minimum value, we need to find the vertex of the parabola.

Finding the Vertex

Using the vertex formula, we can find the x-coordinate of the vertex:

h=−b2a{ h = -\frac{b}{2a} }

h=−−122(3){ h = -\frac{-12}{2(3)} }

h=2{ h = 2 }

Now that we have found the x-coordinate of the vertex, we can find the y-coordinate by substituting the value of x into the quadratic function:

y=3(2)2−12(2)+1{ y = 3(2)^2 - 12(2) + 1 }

y=3(4)−24+1{ y = 3(4) - 24 + 1 }

y=12−24+1{ y = 12 - 24 + 1 }

y=−11{ y = -11 }

Therefore, the minimum value of the function is -11.

Conclusion

In this article, we have discussed how to find the minimum value of a quadratic function. We have also provided an example of how to find the minimum value of a quadratic function using the vertex formula. The minimum value of a quadratic function is the value of the function at the vertex of the parabola, and it can be found using the vertex formula.

Key Takeaways

  • A quadratic function is a polynomial function of degree two.
  • The general form of a quadratic function is given by y = ax^2 + bx + c.
  • The graph of a quadratic function is a parabola that can open upwards or downwards.
  • The minimum value of a quadratic function is the value of the function at the vertex of the parabola.
  • The vertex of a parabola can be found using the vertex formula h = -b/(2a).
  • The y-coordinate of the vertex can be found by substituting the value of x into the quadratic function.

Further Reading

If you want to learn more about quadratic functions and how to find the minimum value of a quadratic function, I recommend checking out the following resources:

  • Khan Academy: Quadratic Functions
  • Mathway: Quadratic Functions
  • Wolfram Alpha: Quadratic Functions

I hope this article has been helpful in understanding how to find the minimum value of a quadratic function. If you have any questions or need further clarification, please don't hesitate to ask.

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Introduction

In our previous article, we discussed how to find the minimum value of a quadratic function. In this article, we will answer some frequently asked questions about quadratic functions and their minimum values.

Q&A

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 given by:

y=ax2+bx+c{ y = ax^2 + bx + c }

where a, b, and c are constants, and x is the variable.

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

A: To find the minimum value of a quadratic function, you need to find the vertex of the parabola. The vertex of a parabola can be found using the vertex formula:

h=−b2a{ h = -\frac{b}{2a} }

where h is the x-coordinate of the vertex, and a and b are the coefficients of the quadratic function.

Q: What is the vertex formula?

A: The vertex formula is a mathematical formula used to find the x-coordinate of the vertex of a parabola. The vertex formula is given by:

h=−b2a{ h = -\frac{b}{2a} }

where h is the x-coordinate of the vertex, and a and b are the coefficients of the quadratic function.

Q: How do I find the y-coordinate of the vertex?

A: To find the y-coordinate of the vertex, you need to substitute the value of x into the quadratic function. The y-coordinate of the vertex can be found using the following formula:

y=a(x−h)2+k{ y = a(x - h)^2 + k }

where y is the y-coordinate of the vertex, a and b are the coefficients of the quadratic function, and h is the x-coordinate of the vertex.

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is the point where the parabola changes direction. The vertex is also the minimum or maximum point of the parabola, depending on whether the parabola opens upwards or downwards.

Q: Can a quadratic function have more than one minimum value?

A: No, a quadratic function can only have one minimum value. The minimum value of a quadratic function is the value of the function at the vertex of the parabola.

Q: Can a quadratic function have no minimum value?

A: Yes, a quadratic function can have no minimum value. This occurs when the parabola opens upwards and has no vertex.

Q: How do I determine whether a quadratic function has a minimum value or not?

A: To determine whether a quadratic function has a minimum value or not, you need to examine the coefficient of the quadratic term. If the coefficient is positive, the parabola opens upwards and has a minimum value. If the coefficient is negative, the parabola opens downwards and has a maximum value.

Q: Can a quadratic function have a maximum value?

A: Yes, a quadratic function can have a maximum value. This occurs when the parabola opens downwards and has a vertex.

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

A: To find the maximum value of a quadratic function, you need to find the vertex of the parabola. The vertex of a parabola can be found using the vertex formula:

h=−b2a{ h = -\frac{b}{2a} }

where h is the x-coordinate of the vertex, and a and b are the coefficients of the quadratic function.

Conclusion

In this article, we have answered some frequently asked questions about quadratic functions and their minimum values. We hope that this article has been helpful in understanding quadratic functions and their minimum values.

Key Takeaways

  • A quadratic function is a polynomial function of degree two.
  • The general form of a quadratic function is given by y = ax^2 + bx + c.
  • The vertex of a parabola can be found using the vertex formula h = -b/(2a).
  • The y-coordinate of the vertex can be found by substituting the value of x into the quadratic function.
  • A quadratic function can only have one minimum value.
  • A quadratic function can have no minimum value if the parabola opens upwards and has no vertex.
  • A quadratic function can have a maximum value if the parabola opens downwards and has a vertex.

Further Reading

If you want to learn more about quadratic functions and their minimum values, I recommend checking out the following resources:

  • Khan Academy: Quadratic Functions
  • Mathway: Quadratic Functions
  • Wolfram Alpha: Quadratic Functions

I hope this article has been helpful in understanding quadratic functions and their minimum values. If you have any questions or need further clarification, please don't hesitate to ask.