Which Values For $h$ And $k$ Are Used To Write The Function $f(x)=x^2+12x+6$ In Vertex Form?A. $h=6, K=36$B. $h=-6, K=-36$C. $h=6, K=30$D. $h=-6, K=-30$

Understanding the Vertex Form of a Quadratic Function

The vertex form of a quadratic function is a way to express a quadratic function in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This form is useful for identifying the vertex of the parabola and for graphing the function.

Converting a Quadratic Function to Vertex Form

To convert a quadratic function to vertex form, we need to complete the square. The general form of a quadratic function is $f(x) = ax^2 + bx + c$. To convert this to vertex form, we need to rewrite the function as $f(x) = a(x - h)^2 + k$, where $h$ and $k$ are constants.

Finding the Values of $h$ and $k$

To find the values of $h$ and $k$, we need to complete the square. We start by factoring out the coefficient of $x^2$, which is $a$. We then take half of the coefficient of $x$ and square it. This gives us the value of $h$. We then substitute this value of $h$ into the function and simplify.

Example: Converting the Function $f(x) = x^2 + 12x + 6$ to Vertex Form

Let's consider the function $f(x) = x^2 + 12x + 6$. To convert this to vertex form, we need to complete the square.

Step 1: Factor out the Coefficient of $x^2$

We start by factoring out the coefficient of $x^2$, which is 1.

$f(x) = x^2 + 12x + 6$

Step 2: Take Half of the Coefficient of $x$ and Square it

We take half of the coefficient of $x$, which is 12, and square it. This gives us the value of $h$.

$h = \left(\frac{12}{2}\right)^2 = 6^2 = 36$

Step 3: Substitute the Value of $h$ into the Function and Simplify

We substitute the value of $h$ into the function and simplify.

$f(x) = (x + 6)^2 + 6 - 36$

$f(x) = (x + 6)^2 - 30$

Step 4: Identify the Values of $h$ and $k$

We can now identify the values of $h$ and $k$.

$h = -6$

$k = -30$

Conclusion

In conclusion, the values of $h$ and $k$ used to write the function $f(x) = x^2 + 12x + 6$ in vertex form are $h = -6$ and $k = -30$. Therefore, the correct answer is:

D. $h = -6, k = -30$

Discussion

This problem requires the student to understand the concept of completing the square and converting a quadratic function to vertex form. The student needs to be able to identify the values of $h$ and $k$ and write the function in vertex form.

Tips and Tricks

• Make sure to factor out the coefficient of $x^2$ before completing the square.
• Take half of the coefficient of $x$ and square it to find the value of $h$.
• Substitute the value of $h$ into the function and simplify to find the value of $k$.
• Use the correct signs for the values of $h$ and $k$.

Practice Problems

1. Convert the function $f(x) = x^2 + 10x + 25$ to vertex form.
2. Convert the function $f(x) = x^2 - 8x + 12$ to vertex form.
3. Convert the function $f(x) = x^2 + 14x + 49$ to vertex form.

Answer Key

1. $f(x) = (x + 5)^2 - 25$
2. $f(x) = (x - 4)^2 + 4$
3. $f(x) = (x + 7)^2 - 36$
   Vertex Form of a Quadratic Function: Q&A =============================================

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is a way to express a quadratic function in the form $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

Q: How do I convert a quadratic function to vertex form?

A: To convert a quadratic function to vertex form, you need to complete the square. You start by factoring out the coefficient of $x^2$, then take half of the coefficient of $x$ and square it. This gives you the value of $h$. You then substitute this value of $h$ into the function and simplify.

Q: What is the significance of the values of $h$ and $k$ in the vertex form of a quadratic function?

A: The values of $h$ and $k$ represent the coordinates of the vertex of the parabola. The value of $h$ represents the x-coordinate of the vertex, and the value of $k$ represents the y-coordinate of the vertex.

Q: How do I find the values of $h$ and $k$ in the vertex form of a quadratic function?

A: To find the values of $h$ and $k$, you need to complete the square. You take half of the coefficient of $x$ and square it to find the value of $h$. You then substitute this value of $h$ into the function and simplify to find the value of $k$.

Q: What is the difference between the standard form and the vertex form of a quadratic function?

A: The standard form of a quadratic function is $f(x) = ax^2 + bx + c$, while the vertex form is $f(x) = a(x - h)^2 + k$. The vertex form is more useful for graphing the function and identifying the vertex of the parabola.

Q: Can I use the vertex form of a quadratic function to find the x-intercepts of the parabola?

A: Yes, you can use the vertex form of a quadratic function to find the x-intercepts of the parabola. The x-intercepts are the points where the parabola intersects the x-axis, and they can be found by setting $y = 0$ in the vertex form of the function.

Q: How do I use the vertex form of a quadratic function to graph the parabola?

A: To graph the parabola using the vertex form, you need to identify the vertex of the parabola, which is represented by the values of $h$ and $k$. You can then use this information to graph the parabola.

Q: Can I use the vertex form of a quadratic function to find the equation of the axis of symmetry?

A: Yes, you can use the vertex form of a quadratic function to find the equation of the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex of the parabola, and it can be found by setting $x = h$ in the vertex form of the function.

Q: How do I use the vertex form of a quadratic function to find the equation of the parabola?

A: To find the equation of the parabola using the vertex form, you need to substitute the values of $h$ and $k$ into the vertex form of the function. This will give you the equation of the parabola in the form $f(x) = a(x - h)^2 + k$.

Q: Can I use the vertex form of a quadratic function to find the maximum or minimum value of the parabola?

A: Yes, you can use the vertex form of a quadratic function to find the maximum or minimum value of the parabola. The maximum or minimum value is represented by the value of $k$ in the vertex form of the function.

Q: How do I use the vertex form of a quadratic function to find the x-coordinate of the vertex?

A: To find the x-coordinate of the vertex using the vertex form, you need to set $y = 0$ in the vertex form of the function and solve for $x$. This will give you the x-coordinate of the vertex.

Q: Can I use the vertex form of a quadratic function to find the y-coordinate of the vertex?

A: Yes, you can use the vertex form of a quadratic function to find the y-coordinate of the vertex. The y-coordinate of the vertex is represented by the value of $k$ in the vertex form of the function.

Q: How do I use the vertex form of a quadratic function to find the equation of the tangent line to the parabola?

A: To find the equation of the tangent line to the parabola using the vertex form, you need to find the slope of the tangent line at the point of tangency. This can be done by finding the derivative of the vertex form of the function and evaluating it at the point of tangency.

Q: Can I use the vertex form of a quadratic function to find the equation of the normal line to the parabola?

A: Yes, you can use the vertex form of a quadratic function to find the equation of the normal line to the parabola. The normal line is a line that is perpendicular to the tangent line at the point of tangency, and it can be found by finding the slope of the normal line and using it to write the equation of the line.

Q: How do I use the vertex form of a quadratic function to find the equation of the asymptote of the parabola?

A: To find the equation of the asymptote of the parabola using the vertex form, you need to find the equation of the line that is parallel to the x-axis and passes through the vertex of the parabola. This can be done by setting $y = 0$ in the vertex form of the function and solving for $x$.