Find The Vertex Of This Quadratic Function.$y = -x^2 - 6x + 15$

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Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding how to find their vertex is crucial for various applications in algebra, calculus, and other fields. In this article, we will delve into the process of finding the vertex of a quadratic function, using the given example: y=−x2−6x+15y = -x^2 - 6x + 15. We will break down the steps involved and provide a clear explanation of each concept.

What is a Quadratic Function?

A quadratic function is a polynomial function of degree two, which means the highest power of the variable (in this case, xx) is two. The general form of a quadratic function is:

y=ax2+bx+cy = ax^2 + bx + c

where aa, bb, and cc are constants, and aa cannot be zero.

The Vertex of a Quadratic Function

The vertex of a quadratic function is the maximum or minimum point of the parabola represented by the function. It is the point where the function changes from increasing to decreasing or vice versa. The vertex is denoted by the coordinates (h,k)(h, k), where hh is the x-coordinate and kk is the y-coordinate.

Finding the Vertex: The Formula

To find the vertex of a quadratic function, we can use the formula:

h=−b2ah = -\frac{b}{2a}

This formula gives us the x-coordinate of the vertex. Once we have the x-coordinate, we can substitute it into the original function to find the y-coordinate.

Applying the Formula to the Given Function

Now, let's apply the formula to the given function: y=−x2−6x+15y = -x^2 - 6x + 15. We have:

a=−1a = -1

b=−6b = -6

Substituting these values into the formula, we get:

h=−−62(−1)h = -\frac{-6}{2(-1)}

h=−−6−2h = -\frac{-6}{-2}

h=−3h = -3

Finding the Y-Coordinate of the Vertex

Now that we have the x-coordinate of the vertex, we can substitute it into the original function to find the y-coordinate:

y=−(−3)2−6(−3)+15y = -(-3)^2 - 6(-3) + 15

y=−9+18+15y = -9 + 18 + 15

y=24y = 24

Conclusion

In this article, we have learned how to find the vertex of a quadratic function using the formula h=−b2ah = -\frac{b}{2a}. We applied this formula to the given function y=−x2−6x+15y = -x^2 - 6x + 15 and found the vertex to be at the point (−3,24)(-3, 24). Understanding how to find the vertex of a quadratic function is essential for various applications in mathematics and other fields.

Example Problems

Problem 1

Find the vertex of the quadratic function y=2x2+5x−3y = 2x^2 + 5x - 3.

Solution

To find the vertex, we need to use the formula h=−b2ah = -\frac{b}{2a}. We have:

a=2a = 2

b=5b = 5

Substituting these values into the formula, we get:

h=−52(2)h = -\frac{5}{2(2)}

h=−54h = -\frac{5}{4}

Now, we can substitute this value into the original function to find the y-coordinate:

y=2(−54)2+5(−54)−3y = 2\left(-\frac{5}{4}\right)^2 + 5\left(-\frac{5}{4}\right) - 3

y=2(2516)−254−3y = 2\left(\frac{25}{16}\right) - \frac{25}{4} - 3

y=258−254−3y = \frac{25}{8} - \frac{25}{4} - 3

y=258−508−248y = \frac{25}{8} - \frac{50}{8} - \frac{24}{8}

y=−498y = -\frac{49}{8}

The vertex of the quadratic function is at the point (−54,−498)\left(-\frac{5}{4}, -\frac{49}{8}\right).

Problem 2

Find the vertex of the quadratic function y=−x2+4x−2y = -x^2 + 4x - 2.

Solution

To find the vertex, we need to use the formula h=−b2ah = -\frac{b}{2a}. We have:

a=−1a = -1

b=4b = 4

Substituting these values into the formula, we get:

h=−42(−1)h = -\frac{4}{2(-1)}

h=−4−2h = -\frac{4}{-2}

h=2h = 2

Now, we can substitute this value into the original function to find the y-coordinate:

y=−(2)2+4(2)−2y = -(2)^2 + 4(2) - 2

y=−4+8−2y = -4 + 8 - 2

y=2y = 2

The vertex of the quadratic function is at the point (2,2)(2, 2).

Tips and Tricks

  • When using the formula h=−b2ah = -\frac{b}{2a}, make sure to substitute the correct values for aa and bb.
  • When finding the y-coordinate of the vertex, substitute the x-coordinate into the original function.
  • The vertex of a quadratic function can be a maximum or minimum point, depending on the value of aa.

Conclusion

In this article, we have learned how to find the vertex of a quadratic function using the formula h=−b2ah = -\frac{b}{2a}. We applied this formula to two example problems and found the vertices of the quadratic functions. Understanding how to find the vertex of a quadratic function is essential for various applications in mathematics and other fields.

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Introduction

In our previous article, we discussed how to find the vertex of a quadratic function using the formula h=−b2ah = -\frac{b}{2a}. However, we understand that there may be some questions and doubts that readers may have. In this article, we will address some of the frequently asked questions related to finding the vertex of a quadratic function.

Q&A

Q1: What is the vertex of a quadratic function?

A1: The vertex of a quadratic function is the maximum or minimum point of the parabola represented by the function. It is the point where the function changes from increasing to decreasing or vice versa.

Q2: How do I find the vertex of a quadratic function?

A2: To find the vertex of a quadratic function, you can use the formula h=−b2ah = -\frac{b}{2a}. This formula gives you the x-coordinate of the vertex. Once you have the x-coordinate, you can substitute it into the original function to find the y-coordinate.

Q3: What is the significance of the vertex of a quadratic function?

A3: The vertex of a quadratic function is significant because it represents the maximum or minimum point of the parabola. This point is crucial in various applications, such as optimization problems, physics, and engineering.

Q4: Can the vertex of a quadratic function be a maximum or minimum point?

A4: Yes, the vertex of a quadratic function can be a maximum or minimum point, depending on the value of aa. If aa is positive, the vertex is a minimum point. If aa is negative, the vertex is a maximum point.

Q5: How do I determine if the vertex of a quadratic function is a maximum or minimum point?

A5: To determine if the vertex of a quadratic function is a maximum or minimum point, you can look at the value of aa. If aa is positive, the vertex is a minimum point. If aa is negative, the vertex is a maximum point.

Q6: Can the vertex of a quadratic function be a point of inflection?

A6: No, the vertex of a quadratic function cannot be a point of inflection. A point of inflection is a point where the concavity of the function changes, but the vertex is a point where the function changes from increasing to decreasing or vice versa.

Q7: How do I find the vertex of a quadratic function with a negative leading coefficient?

A7: To find the vertex of a quadratic function with a negative leading coefficient, you can use the formula h=−b2ah = -\frac{b}{2a}. This formula gives you the x-coordinate of the vertex. Once you have the x-coordinate, you can substitute it into the original function to find the y-coordinate.

Q8: Can the vertex of a quadratic function be a point of discontinuity?

A8: No, the vertex of a quadratic function cannot be a point of discontinuity. A point of discontinuity is a point where the function is not defined, but the vertex is a point where the function is defined.

Q9: How do I find the vertex of a quadratic function with a fractional coefficient?

A9: To find the vertex of a quadratic function with a fractional coefficient, you can use the formula h=−b2ah = -\frac{b}{2a}. This formula gives you the x-coordinate of the vertex. Once you have the x-coordinate, you can substitute it into the original function to find the y-coordinate.

Q10: Can the vertex of a quadratic function be a point of tangency?

A10: No, the vertex of a quadratic function cannot be a point of tangency. A point of tangency is a point where the function is tangent to a line, but the vertex is a point where the function changes from increasing to decreasing or vice versa.

Conclusion

In this article, we have addressed some of the frequently asked questions related to finding the vertex of a quadratic function. We hope that this article has provided you with a better understanding of the concept of the vertex of a quadratic function and how to find it. If you have any further questions or doubts, please feel free to ask.

Tips and Tricks

  • When using the formula h=−b2ah = -\frac{b}{2a}, make sure to substitute the correct values for aa and bb.
  • When finding the y-coordinate of the vertex, substitute the x-coordinate into the original function.
  • The vertex of a quadratic function can be a maximum or minimum point, depending on the value of aa.
  • The vertex of a quadratic function cannot be a point of inflection, discontinuity, or tangency.

Example Problems

Problem 1

Find the vertex of the quadratic function y=2x2+5x−3y = 2x^2 + 5x - 3.

Solution

To find the vertex, we need to use the formula h=−b2ah = -\frac{b}{2a}. We have:

a=2a = 2

b=5b = 5

Substituting these values into the formula, we get:

h=−52(2)h = -\frac{5}{2(2)}

h=−54h = -\frac{5}{4}

Now, we can substitute this value into the original function to find the y-coordinate:

y=2(−54)2+5(−54)−3y = 2\left(-\frac{5}{4}\right)^2 + 5\left(-\frac{5}{4}\right) - 3

y=2(2516)−254−3y = 2\left(\frac{25}{16}\right) - \frac{25}{4} - 3

y=258−508−248y = \frac{25}{8} - \frac{50}{8} - \frac{24}{8}

y=−498y = -\frac{49}{8}

The vertex of the quadratic function is at the point (−54,−498)\left(-\frac{5}{4}, -\frac{49}{8}\right).

Problem 2

Find the vertex of the quadratic function y=−x2+4x−2y = -x^2 + 4x - 2.

Solution

To find the vertex, we need to use the formula h=−b2ah = -\frac{b}{2a}. We have:

a=−1a = -1

b=4b = 4

Substituting these values into the formula, we get:

h=−42(−1)h = -\frac{4}{2(-1)}

h=−4−2h = -\frac{4}{-2}

h=2h = 2

Now, we can substitute this value into the original function to find the y-coordinate:

y=−(2)2+4(2)−2y = -(2)^2 + 4(2) - 2

y=−4+8−2y = -4 + 8 - 2

y=2y = 2

The vertex of the quadratic function is at the point (2,2)(2, 2).

Conclusion

In this article, we have provided answers to some of the frequently asked questions related to finding the vertex of a quadratic function. We hope that this article has provided you with a better understanding of the concept of the vertex of a quadratic function and how to find it. If you have any further questions or doubts, please feel free to ask.