Find The Vertex Of: $f(x) = -x^2 + 6x - 12$.The Vertex Is \[$\square\$\].

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 $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The vertex of a quadratic function is the maximum or minimum point of the parabola it represents. In this article, we will focus on finding the vertex of the quadratic function $f(x) = -x^2 + 6x - 12$.

Understanding the Vertex Formula

The vertex of a quadratic function can be found using the formula $\left( -\frac{b}{2a}, f\left( -\frac{b}{2a} \right) \right)$. This formula gives us the x-coordinate and the corresponding y-coordinate of the vertex. To find the vertex, we need to identify the values of $a$ and $b$ in the quadratic function.

Identifying the Values of $a$ and $b$

In the given quadratic function $f(x) = -x^2 + 6x - 12$, we can see that $a = -1$ and $b = 6$. Now that we have identified the values of $a$ and $b$, we can use the vertex formula to find the vertex of the quadratic function.

Applying the Vertex Formula

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

$$x = -\frac{b}{2a} = -\frac{6}{2(-1)} = -\frac{6}{-2} = 3$$

Now that we have found the x-coordinate of the vertex, we can find the corresponding y-coordinate by substituting $x = 3$ into the quadratic function:

$$f(3) = -(3)^2 + 6(3) - 12 = -9 + 18 - 12 = -3$$

Therefore, the vertex of the quadratic function $f(x) = -x^2 + 6x - 12$ is $\left( 3, -3 \right)$.

Interpretation of the Vertex

The vertex of a quadratic function represents the maximum or minimum point of the parabola it represents. In this case, the vertex $\left( 3, -3 \right)$ represents the minimum point of the parabola. This means that the parabola opens downwards, and the vertex is the lowest point on the parabola.

Conclusion

In this article, we have discussed how to find the vertex of a quadratic function using the vertex formula. We have applied the formula to the quadratic function $f(x) = -x^2 + 6x - 12$ and found the vertex to be $\left( 3, -3 \right)$. The vertex represents the minimum point of the parabola, and it is an important concept in mathematics and science.

Example Problems

1. Find the vertex of the quadratic function $f(x) = x^2 + 4x + 4$.
2. Find the vertex of the quadratic function $f(x) = -2x^2 + 6x - 3$.

Solutions

1. Using the vertex formula, we can find the x-coordinate of the vertex as follows:

$$x = -\frac{b}{2a} = -\frac{4}{2(1)} = -\frac{4}{2} = -2$$

Now that we have found the x-coordinate of the vertex, we can find the corresponding y-coordinate by substituting $x = -2$ into the quadratic function:

$$f(-2) = (-2)^2 + 4(-2) + 4 = 4 - 8 + 4 = 0$$

Therefore, the vertex of the quadratic function $f(x) = x^2 + 4x + 4$ is $\left( -2, 0 \right)$.

2. Using the vertex formula, we can find the x-coordinate of the vertex as follows:

$$x = -\frac{b}{2a} = -\frac{6}{2(-2)} = -\frac{6}{-4} = \frac{3}{2}$$

Now that we have found the x-coordinate of the vertex, we can find the corresponding y-coordinate by substituting $x = \frac{3}{2}$ into the quadratic function:

$$f\left( \frac{3}{2} \right) = -2\left( \frac{3}{2} \right)^2 + 6\left( \frac{3}{2} \right) - 3 = -\frac{9}{2} + 9 - 3 = -\frac{9}{2} + 6 = -\frac{9}{2} + \frac{12}{2} = \frac{3}{2}$$

Therefore, the vertex of the quadratic function $f(x) = -2x^2 + 6x - 3$ is $\left( \frac{3}{2}, \frac{3}{2} \right)$.

Final Thoughts

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 it represents. It is the point where the parabola changes direction, either from increasing to decreasing or from decreasing to increasing.

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

A: To find the vertex of a quadratic function, you can use the vertex formula: $\left( -\frac{b}{2a}, f\left( -\frac{b}{2a} \right) \right)$. This formula gives you the x-coordinate and the corresponding y-coordinate of the vertex.

Q: What are the values of $a$ and $b$ in the vertex formula?

A: The values of $a$ and $b$ are the coefficients of the quadratic function. In the general form of a quadratic function, $f(x) = ax^2 + bx + c$, $a$ is the coefficient of the $x^2$ term and $b$ is the coefficient of the $x$ term.

Q: How do I apply the vertex formula?

A: To apply the vertex formula, you need to substitute the values of $a$ and $b$ into the formula and simplify. Then, you can find the x-coordinate of the vertex by dividing $b$ by $2a$. Finally, you can find the corresponding y-coordinate by substituting the x-coordinate into the quadratic function.

Q: What if the quadratic function has a negative leading coefficient?

A: If the quadratic function has a negative leading coefficient, the parabola opens downwards, and the vertex is the maximum point. In this case, the vertex formula still applies, but the x-coordinate and y-coordinate will be different.

Q: Can I find the vertex of a quadratic function without using the vertex formula?

A: Yes, you can find the vertex of a quadratic function without using the vertex formula. One way to do this is to complete the square, which involves rewriting the quadratic function in the form $f(x) = a(x - h)^2 + k$. The vertex of the parabola is then given by the point $(h, k)$.

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. This point is important in many applications, such as physics, engineering, and economics, where the parabola represents a relationship between two variables.

Q: Can I use the vertex formula to find the vertex of a quadratic function with a complex coefficient?

A: Yes, you can use the vertex formula to find the vertex of a quadratic function with a complex coefficient. However, you need to be careful when simplifying the expression and finding the x-coordinate and y-coordinate of the vertex.

Q: Are there any other ways to find the vertex of a quadratic function?

A: Yes, there are other ways to find the vertex of a quadratic function, such as using the graphing calculator or the online vertex calculator. However, the vertex formula is a simple and efficient way to find the vertex of a quadratic function.

Q: Can I find the vertex of a quadratic function with a non-integer coefficient?

A: Yes, you can find the vertex of a quadratic function with a non-integer coefficient. However, you need to be careful when simplifying the expression and finding the x-coordinate and y-coordinate of the vertex.

Q: What if the quadratic function has a repeated root?

A: If the quadratic function has a repeated root, the parabola touches the x-axis at that point, and the vertex is not defined. In this case, the quadratic function can be written in the form $f(x) = a(x - r)^2$, where $r$ is the repeated root.

Q: Can I use the vertex formula to find the vertex of a quadratic function with a rational coefficient?

A: Yes, you can use the vertex formula to find the vertex of a quadratic function with a rational coefficient. However, you need to be careful when simplifying the expression and finding the x-coordinate and y-coordinate of the vertex.

Q: Are there any other applications of the vertex formula?

A: Yes, there are many other applications of the vertex formula, such as finding the maximum or minimum value of a quadratic function, determining the direction of the parabola, and solving quadratic equations.