The Parent Function Of The Function $g(x) = (x-h)^2 + K$ Is $f(x) = X^2$. The Vertex Of The Function $ G ( X ) G(x) G ( X ) [/tex] Is Located At $(9, -8)$.What Are The Values Of $h$ And $ K K K [/tex]?

Understanding the Parent Function

In mathematics, a parent function is a basic function from which other functions can be derived. For quadratic functions, the parent function is $f(x) = x^2$. This function is a quadratic function in its simplest form, and it serves as the foundation for all other quadratic functions.

The Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by $g(x) = (x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This form is useful because it allows us to easily identify the vertex of the parabola, which is the maximum or minimum point of the function.

The Given Function and Its Vertex

We are given the function $g(x) = (x-h)^2 + k$ and told that its vertex is located at $(9, -8)$. This means that the value of $h$ is $9$ and the value of $k$ is $-8$.

Finding the Values of $h$ and $k$

To find the values of $h$ and $k$, we can substitute the values of $h$ and $k$ into the function $g(x) = (x-h)^2 + k$. This gives us:

$$g(x) = (x-9)^2 - 8$$

Expanding the Function

To expand the function, we can use the formula $(a-b)^2 = a^2 - 2ab + b^2$. Applying this formula to the function $g(x) = (x-9)^2 - 8$, we get:

$$g(x) = x^2 - 18x + 81 - 8$$

Simplifying the function, we get:

$$g(x) = x^2 - 18x + 73$$

Comparing the Expanded Function to the Parent Function

Now that we have expanded the function $g(x)$, we can compare it to the parent function $f(x) = x^2$. We can see that the expanded function has a leading coefficient of $1$, which is the same as the parent function.

Conclusion

In conclusion, we have found the values of $h$ and $k$ for the function $g(x) = (x-h)^2 + k$. We have also expanded the function and compared it to the parent function $f(x) = x^2$. The values of $h$ and $k$ are $9$ and $-8$, respectively.

The Final Answer

The final answer is:

$h = 9$ $k = -8$

Additional Information

• The parent function of a quadratic function is $f(x) = x^2$.
• The vertex form of a quadratic function is $g(x) = (x-h)^2 + k$.
• The values of $h$ and $k$ can be found by substituting the values of $h$ and $k$ into the function $g(x) = (x-h)^2 + k$.
• The expanded function can be compared to the parent function to verify the values of $h$ and $k$.
  The Parent Function of a Quadratic Function: Q&A =====================================================

Q: What is the parent function of a quadratic function?

A: The parent function of a quadratic function is $f(x) = x^2$. This function is a quadratic function in its simplest form and serves as the foundation for all other quadratic functions.

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

A: The vertex form of a quadratic function is $g(x) = (x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola. This form is useful because it allows us to easily identify the vertex of the parabola, which is the maximum or minimum point of the function.

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

A: To find the values of $h$ and $k$, you can substitute the values of $h$ and $k$ into the function $g(x) = (x-h)^2 + k$. This will give you the expanded form of the function, which can be compared to the parent function to verify the values of $h$ and $k$.

Q: What is the difference between the parent function and a quadratic function in vertex form?

A: The parent function is a basic quadratic function in its simplest form, while a quadratic function in vertex form is a more general form that includes the vertex of the parabola. The vertex form is useful because it allows us to easily identify the vertex of the parabola, which is the maximum or minimum point of the function.

Q: How do I expand a quadratic function in vertex form?

A: To expand a quadratic function in vertex form, you can use the formula $(a-b)^2 = a^2 - 2ab + b^2$. Applying this formula to the function $g(x) = (x-h)^2 + k$, you get:

$$g(x) = x^2 - 2hx + h^2 + k$$

Q: What is the significance of the vertex of a parabola?

A: The vertex of a parabola is the maximum or minimum point of the function. It is the point at which the function changes from increasing to decreasing or from decreasing to increasing.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you can use the formula $x = -\frac{b}{2a}$, where $a$ and $b$ are the coefficients of the quadratic function. This will give you the x-coordinate of the vertex. To find the y-coordinate of the vertex, you can substitute the x-coordinate into the function.

Q: What is the relationship between the parent function and a quadratic function in vertex form?

A: The parent function is a special case of a quadratic function in vertex form, where $h = 0$ and $k = 0$. This means that the parent function is a quadratic function in its simplest form, with no horizontal or vertical shifts.

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

A: To use the parent function to find the values of $h$ and $k$, you can compare the expanded form of the function to the parent function. This will allow you to identify the values of $h$ and $k$ that are needed to transform the parent function into the given function.

Q: What are some common mistakes to avoid when working with quadratic functions in vertex form?

A: Some common mistakes to avoid when working with quadratic functions in vertex form include:

• Failing to identify the vertex of the parabola
• Failing to expand the function correctly
• Failing to compare the expanded function to the parent function
• Failing to identify the values of $h$ and $k$ that are needed to transform the parent function into the given function.