What Is The Minimum Value Of The Given Function $f(x)=(2x-13)(2x+9)$?A. -121 B. $-\frac{9}{2}$ C. 1 D. 4

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Introduction

In mathematics, a function is a relation between a set of inputs, called the domain, and a set of possible outputs, called the range. The minimum value of a function is the smallest value that the function can take. In this article, we will explore the minimum value of the given function $f(x)=(2x-13)(2x+9)$.

Understanding the Function

The given function is a quadratic function, which is a polynomial function of degree two. It can be written in the form $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are constants. In this case, the function can be written as $f(x)=4x^2-26x-117$.

Finding the Minimum Value

To find the minimum value of the function, we need to find the vertex of the parabola. The vertex of a parabola is the point where the parabola changes direction. It is the minimum or maximum point of the parabola. In this case, we are looking for the minimum value, so we need to find the vertex of the parabola.

Using the Formula for the Vertex

The formula for the vertex of a parabola is $x=-\frac{b}{2a}$. In this case, $a=4$ and $b=-26$. Plugging these values into the formula, we get $x=-\frac{-26}{2(4)}=\frac{26}{8}=\frac{13}{4}$.

Finding the Minimum Value

Now that we have found the x-coordinate of the vertex, we can find the minimum value of the function by plugging this value into the function. We get $f(\frac{13}{4})=4(\frac{13}{4})^2-26(\frac{13}{4})-117$.

Simplifying the Expression

Simplifying the expression, we get $f(\frac{13}{4})=\frac{169}{4}-\frac{338}{4}-\frac{468}{4}=-\frac{637}{4}$.

Conclusion

In conclusion, the minimum value of the given function $f(x)=(2x-13)(2x+9)$ is $-\frac{637}{4}$.

Comparison with the Options

Now, let's compare our answer with the options given in the problem.

  • Option A: -121
  • Option B: $-\frac{9}{2}$
  • Option C: 1
  • Option D: 4

Our answer, $-\frac{637}{4}$, is not equal to any of the options. However, we can simplify it to $-\frac{637}{4}=-\frac{9}{2}\times\frac{71}{2}$.

Final Answer

Therefore, the correct answer is not among the options given in the problem. However, we can see that option B, $-\frac{9}{2}$, is a factor of the minimum value of the function.

Discussion

The minimum value of a function is an important concept in mathematics. It is used to find the minimum or maximum point of a function. In this article, we have seen how to find the minimum value of a quadratic function using the formula for the vertex.

Related Topics

  • Quadratic Functions
  • Vertex of a Parabola
  • Minimum Value of a Function

References

  • [1] "Quadratic Functions" by Math Open Reference
  • [2] "Vertex of a Parabola" by Math Is Fun
  • [3] "Minimum Value of a Function" by Khan Academy

Introduction

In our previous article, we explored the minimum value of the given function $f(x)=(2x-13)(2x+9)$. In this article, we will answer some frequently asked questions related to the minimum value of a function.

Q1: What is the minimum value of a function?

A1: The minimum value of a function is the smallest value that the function can take. It is the point where the function changes direction, and it is the minimum or maximum point of the function.

Q2: How do I find the minimum value of a function?

A2: To find the minimum value of a function, you need to find the vertex of the parabola. The vertex of a parabola is the point where the parabola changes direction. You can use the formula for the vertex, which is $x=-\frac{b}{2a}$, to find the x-coordinate of the vertex.

Q3: What is the formula for the vertex of a parabola?

A3: The formula for the vertex of a parabola is $x=-\frac{b}{2a}$. This formula is used to find the x-coordinate of the vertex of a parabola.

Q4: How do I find the minimum value of a quadratic function?

A4: To find the minimum value of a quadratic function, you need to find the vertex of the parabola. You can use the formula for the vertex, which is $x=-\frac{b}{2a}$, to find the x-coordinate of the vertex. Then, you can plug this value into the function to find the minimum value.

Q5: What is the minimum value of the function $f(x)=x^2-6x+8$?

A5: To find the minimum value of the function $f(x)=x^2-6x+8$, you need to find the vertex of the parabola. Using the formula for the vertex, you get $x=-\frac{-6}{2(1)}=3$. Plugging this value into the function, you get $f(3)=3^2-6(3)+8=1$.

Q6: What is the minimum value of the function $f(x)=2x^2-12x+20$?

A6: To find the minimum value of the function $f(x)=2x^2-12x+20$, you need to find the vertex of the parabola. Using the formula for the vertex, you get $x=-\frac{-12}{2(2)}=3$. Plugging this value into the function, you get $f(3)=2(3)^2-12(3)+20=4$.

Q7: How do I know if a function has a minimum value?

A7: A function has a minimum value if it is a quadratic function and the coefficient of the squared term is positive. If the coefficient of the squared term is negative, the function has a maximum value.

Q8: What is the minimum value of the function $f(x)=-x^2+6x-8$?

A8: To find the minimum value of the function $f(x)=-x^2+6x-8$, you need to find the vertex of the parabola. Using the formula for the vertex, you get $x=-\frac{6}{2(-1)}=-3$. Plugging this value into the function, you get $f(-3)=-(-3)^2+6(-3)-8=1$.

Conclusion

In conclusion, the minimum value of a function is an important concept in mathematics. It is used to find the minimum or maximum point of a function. In this article, we have answered some frequently asked questions related to the minimum value of a function.

Related Topics

  • Quadratic Functions
  • Vertex of a Parabola
  • Minimum Value of a Function

References

  • [1] "Quadratic Functions" by Math Open Reference
  • [2] "Vertex of a Parabola" by Math Is Fun
  • [3] "Minimum Value of a Function" by Khan Academy