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

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 the function $f(x) = x^2$. This function is the simplest form of a quadratic function and 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 the equation $g(x) = (x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. In this equation, $h$ represents the horizontal shift of the parabola, and $k$ represents the vertical shift.

The Given Function and Its Vertex

The given function is $g(x) = (x - h)^2 + k$, and its vertex is located at $(9, -8)$. This means that the function can be written in the form $g(x) = (x - 9)^2 - 8$.

Finding the Values of $h$ and $k$

To find the values of $h$ and $k$, we can compare the given function with the standard form of a quadratic function. The standard form of a quadratic function is $f(x) = ax^2 + bx + c$. By comparing the given function with the standard form, we can see that $a = 1$, $b = 0$, and $c = -8$.

The Relationship Between the Given Function and the Parent Function

The parent function of the given function is $f(x) = x^2$. To find the values of $h$ and $k$, we need to find the values of $a$, $b$, and $c$ in the standard form of the parent function. The standard form of the parent function is $f(x) = x^2$, which can be written as $f(x) = 1x^2 + 0x + 0$.

Comparing the Given Function with the Parent Function

By comparing the given function with the parent function, we can see that the given function is a transformation of the parent function. The transformation involves a horizontal shift of $9$ units to the right and a vertical shift of $8$ units down.

Finding the Values of $h$ and $k$

Since the given function is a transformation of the parent function, we can use the values of $h$ and $k$ to find the values of $a$, $b$, and $c$ in the standard form of the given function. The values of $h$ and $k$ are given by the vertex of the parabola, which is $(9, -8)$. Therefore, the values of $h$ and $k$ are $h = 9$ and $k = -8$.

Conclusion

In conclusion, the parent function of the given function is $f(x) = x^2$. The vertex of the given function is located at $(9, -8)$, which means that the values of $h$ and $k$ are $h = 9$ and $k = -8$. These values can be used to find the values of $a$, $b$, and $c$ in the standard form of the given function.

The Standard Form of a Quadratic Function

A quadratic function can be written in the standard form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The standard form of a quadratic function is useful for finding the vertex of the parabola and for graphing the function.

The Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by the equation $g(x) = (x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The vertex form of a quadratic function is useful for finding the vertex of the parabola and for graphing the function.

The Parent Function of a Quadratic Function

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

The Relationship Between the Given Function and the Parent Function

The given function is a transformation of the parent function. The transformation involves a horizontal shift of $9$ units to the right and a vertical shift of $8$ units down.

Finding the Values of $h$ and $k$

The values of $h$ and $k$ are given by the vertex of the parabola, which is $(9, -8)$. Therefore, the values of $h$ and $k$ are $h = 9$ and $k = -8$.

Conclusion

In conclusion, the parent function of the given function is $f(x) = x^2$. The vertex of the given function is located at $(9, -8)$, which means that the values of $h$ and $k$ are $h = 9$ and $k = -8$. These values can be used to find the values of $a$, $b$, and $c$ in the standard form of the given function.

The Standard Form of a Quadratic Function

A quadratic function can be written in the standard form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. The standard form of a quadratic function is useful for finding the vertex of the parabola and for graphing the function.

The Vertex Form of a Quadratic Function

The vertex form of a quadratic function is given by the equation $g(x) = (x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The vertex form of a quadratic function is useful for finding the vertex of the parabola and for graphing the function.

The Parent Function of a Quadratic Function

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

The Relationship Between the Given Function and the Parent Function

The given function is a transformation of the parent function. The transformation involves a horizontal shift of $9$ units to the right and a vertical shift of $8$ units down.

Finding the Values of $h$ and $k$

The values of $h$ and $k$ are given by the vertex of the parabola, which is $(9, -8)$. Therefore, the values of $h$ and $k$ are $h = 9$ and $k = -8$.

Conclusion

$$$$$$$$$$$$$$$$$$

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

A: The parent function of a quadratic function is the function $f(x) = x^2$. This function is the simplest form of a quadratic function 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 given by the equation $g(x) = (x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The vertex form of a quadratic function is useful for finding the vertex of the parabola and for graphing the function.

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 find the vertex of the parabola. The vertex of the parabola is given by the coordinates $(h, k)$. You can find the vertex by using the formula $h = -\frac{b}{2a}$ and $k = f(h)$.

Q: What is the relationship between the given function and the parent function?

A: The given function is a transformation of the parent function. The transformation involves a horizontal shift of $9$ units to the right and a vertical shift of $8$ units down.

Q: How do I find the values of $a$, $b$, and $c$ in the standard form of a quadratic function?

A: To find the values of $a$, $b$, and $c$, you need to compare the given function with the standard form of a quadratic function. The standard form of a quadratic function is $f(x) = ax^2 + bx + c$. You can find the values of $a$, $b$, and $c$ by comparing the given function with the standard form.

Q: What is the significance of the parent function in mathematics?

A: The parent function is the simplest form of a quadratic function and serves as the foundation for all other quadratic functions. It is used as a reference point to find the values of $a$, $b$, and $c$ in the standard form of a quadratic function.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you need to find the vertex of the parabola and the values of $a$, $b$, and $c$. You can use the vertex form of a quadratic function to find the vertex and the values of $a$, $b$, and $c$.

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

A: The vertex form of a quadratic function is given by the equation $g(x) = (x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola. The standard form of a quadratic function is $f(x) = ax^2 + bx + c$. The vertex form is useful for finding the vertex of the parabola and for graphing the function, while the standard form is useful for finding the values of $a$, $b$, and $c$.

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

A: To find the vertex of a parabola, you need to find the values of $h$ and $k$ in the vertex form of a quadratic function. The vertex of the parabola is given by the coordinates $(h, k)$.

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

A: The vertex of a parabola is the highest or lowest point of the parabola. It is used as a reference point to find the values of $a$, $b$, and $c$ in the standard form of a quadratic function.

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

A: To use the vertex form of a quadratic function to graph a parabola, you need to find the vertex of the parabola and the values of $a$, $b$, and $c$. You can use the vertex form to find the vertex and the values of $a$, $b$, and $c$, and then graph the parabola using the vertex and the values of $a$, $b$, and $c$.

Q: What is the difference between a quadratic function and a linear function?

A: A quadratic function is a function of the form $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. A linear function is a function of the form $f(x) = mx + b$, where $m$ and $b$ are constants. The main difference between a quadratic function and a linear function is that a quadratic function has a parabolic shape, while a linear function has a straight line shape.

Q: How do I determine whether a function is quadratic or linear?

A: To determine whether a function is quadratic or linear, you need to look at the form of the function. If the function is of the form $f(x) = ax^2 + bx + c$, it is a quadratic function. If the function is of the form $f(x) = mx + b$, it is a linear function.

Q: What is the significance of the quadratic formula?

A: The quadratic formula is a formula used to find the solutions to a quadratic equation. It is given by the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. The quadratic formula is used to find the solutions to a quadratic equation and is an important tool in algebra.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to plug in the values of $a$, $b$, and $c$ into the formula. The formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. You can then simplify the expression to find the solutions to the quadratic equation.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. A linear equation is an equation of the form $mx + b = 0$, where $m$ and $b$ are constants. The main difference between a quadratic equation and a linear equation is that a quadratic equation has a parabolic shape, while a linear equation has a straight line shape.

Q: How do I determine whether an equation is quadratic or linear?

A: To determine whether an equation is quadratic or linear, you need to look at the form of the equation. If the equation is of the form $ax^2 + bx + c = 0$, it is a quadratic equation. If the equation is of the form $mx + b = 0$, it is a linear equation.