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Introduction

In mathematics, functions are used to describe the relationship between variables. A function can be represented in various forms, including factored form, which is a way of expressing a function as a product of its factors. In this article, we will explore how to determine the minimum value of a function with zeros located at -1 and 5, and a y-intercept of (0, -25).

Understanding the Function

A function in factored form can be written as:

f(x) = a(x - r1)(x - r2)

where a is the leading coefficient, and r1 and r2 are the zeros of the function. In this case, we are given that the zeros are located at -1 and 5, so we can write the function as:

f(x) = a(x + 1)(x - 5)

We are also given that the y-intercept is (0, -25), which means that when x = 0, f(x) = -25. We can use this information to find the value of a.

Finding the Value of a

To find the value of a, we can substitute x = 0 into the function and set it equal to -25:

f(0) = a(0 + 1)(0 - 5) -25 = a(-5) a = 5

Now that we have found the value of a, we can write the function as:

f(x) = 5(x + 1)(x - 5)

Determining the Minimum Value

To determine the minimum value of the function, we need to find the vertex of the parabola. The vertex is the point on the parabola where the function changes from decreasing to increasing. The x-coordinate of the vertex can be found using the formula:

x = (-b / 2a)

In this case, a = 5 and b = 0 (since there is no linear term in the function), so:

x = (0 / 2(5)) x = 0

Now that we have found the x-coordinate of the vertex, we can find the y-coordinate by substituting x = 0 into the function:

f(0) = 5(0 + 1)(0 - 5) f(0) = -25

Therefore, the minimum value of the function is -25, which occurs at x = 0.

Conclusion

In this article, we have seen how to determine the minimum value of a function with zeros located at -1 and 5, and a y-intercept of (0, -25). We have used the factored form of the function to find the value of the leading coefficient, and then used the vertex formula to find the x-coordinate of the vertex. Finally, we have substituted the x-coordinate into the function to find the y-coordinate of the vertex, which is the minimum value of the function.

Key Takeaways

  • A function in factored form can be written as f(x) = a(x - r1)(x - r2), where a is the leading coefficient, and r1 and r2 are the zeros of the function.
  • The y-intercept of a function is the point on the y-axis where the function crosses the y-axis.
  • The vertex of a parabola is the point on the parabola where the function changes from decreasing to increasing.
  • The x-coordinate of the vertex can be found using the formula x = (-b / 2a).
  • The y-coordinate of the vertex can be found by substituting the x-coordinate into the function.

Further Reading

For more information on functions and their properties, see the following resources:

References

Introduction

In our previous article, we explored how to determine the minimum value of a function with zeros located at -1 and 5, and a y-intercept of (0, -25). We used the factored form of the function to find the value of the leading coefficient, and then used the vertex formula to find the x-coordinate of the vertex. Finally, we substituted the x-coordinate into the function to find the y-coordinate of the vertex, which is the minimum value of the function.

Q&A

Q: What is the factored form of a function? A: The factored form of a function is a way of expressing a function as a product of its factors. It is written as f(x) = a(x - r1)(x - r2), where a is the leading coefficient, and r1 and r2 are the zeros of the function.

Q: How do I find the value of the leading coefficient? A: To find the value of the leading coefficient, you can substitute x = 0 into the function and set it equal to the y-intercept. This will give you an equation that you can solve for the leading coefficient.

Q: What is the vertex of a parabola? A: The vertex of a parabola is the point on the parabola where the function changes from decreasing to increasing. It is the minimum or maximum point of the parabola.

Q: How do I find the x-coordinate of the vertex? A: To find the x-coordinate of the vertex, you can use the formula x = (-b / 2a). This formula is derived from the factored form of the function.

Q: How do I find the y-coordinate of the vertex? A: To find the y-coordinate of the vertex, you can substitute the x-coordinate into the function. This will give you the minimum or maximum value of the function.

Q: What is the minimum value of the function? A: The minimum value of the function is the y-coordinate of the vertex. It is the lowest point on the parabola.

Q: How do I use the factored form of a function to find the minimum value? A: To use the factored form of a function to find the minimum value, you need to follow these steps:

  1. Write the function in factored form: f(x) = a(x - r1)(x - r2)
  2. Find the value of the leading coefficient by substituting x = 0 into the function and setting it equal to the y-intercept.
  3. Find the x-coordinate of the vertex using the formula x = (-b / 2a).
  4. Find the y-coordinate of the vertex by substituting the x-coordinate into the function.

Q: What are some common mistakes to avoid when finding the minimum value of a function? A: Some common mistakes to avoid when finding the minimum value of a function include:

  • Not writing the function in factored form
  • Not finding the value of the leading coefficient
  • Not using the correct formula to find the x-coordinate of the vertex
  • Not substituting the x-coordinate into the function to find the y-coordinate of the vertex

Conclusion

In this article, we have answered some common questions about finding the minimum value of a function. We have covered the factored form of a function, the vertex of a parabola, and the steps to follow to find the minimum value of a function. We hope that this article has been helpful in clarifying any confusion you may have had about finding the minimum value of a function.

Key Takeaways

  • The factored form of a function is a way of expressing a function as a product of its factors.
  • The vertex of a parabola is the point on the parabola where the function changes from decreasing to increasing.
  • The x-coordinate of the vertex can be found using the formula x = (-b / 2a).
  • The y-coordinate of the vertex can be found by substituting the x-coordinate into the function.
  • The minimum value of a function is the y-coordinate of the vertex.

Further Reading

For more information on functions and their properties, see the following resources:

References