The Function G ( X ) = 10 X 2 − 100 X + 213 G(x) = 10x^2 - 100x + 213 G ( X ) = 10 X 2 − 100 X + 213 Written In Vertex Form Is G ( X ) = 10 ( X − 5 ) 2 − 37 G(x) = 10(x-5)^2 - 37 G ( X ) = 10 ( X − 5 ) 2 − 37 . Which Statements Are True About G ( X G(x G ( X ]? Select Three Options.A. The Axis Of Symmetry Is The Line X = − 5 X = -5 X = − 5 .B. The Vertex Of

Introduction

In mathematics, the vertex form of a quadratic function is a powerful tool for analyzing and graphing quadratic equations. The vertex form of a quadratic function is given by $g(x) = a(x-h)^2 + k$, where $(h,k)$ is the vertex of the parabola. In this article, we will explore the properties of the function $g(x) = 10x^2 - 100x + 213$ when written in vertex form as $g(x) = 10(x-5)^2 - 37$. We will examine three statements about $g(x)$ and determine which ones are true.

The Vertex Form of $g(x)$

The vertex form of a quadratic function is given by $g(x) = a(x-h)^2 + k$. To convert the function $g(x) = 10x^2 - 100x + 213$ to vertex form, we need to complete the square. We can do this by first factoring out the coefficient of the $x^2$ term, which is 10.

import sympy as sp
x = sp.symbols('x')

g = 10x**2 - 100x + 213

g = 10*(x**2 - 10*x) + 213

Next, we need to add and subtract the square of half the coefficient of the $x$ term inside the parentheses. The coefficient of the $x$ term is -10, so half of this is -5, and the square of -5 is 25.

# Add and subtract the square of half the coefficient of the x term
g = 10*(x**2 - 10*x + 25 - 25) + 213

Now, we can rewrite the expression inside the parentheses as a perfect square trinomial.

# Rewrite the expression inside the parentheses as a perfect square trinomial
g = 10*((x - 5)**2 - 25) + 213

Finally, we can simplify the expression by distributing the 10 to the terms inside the parentheses.

# Simplify the expression by distributing the 10 to the terms inside the parentheses
g = 10*(x - 5)**2 - 250 + 213

Therefore, the vertex form of the function $g(x) = 10x^2 - 100x + 213$ is $g(x) = 10(x-5)^2 - 37$.

Statement A: The Axis of Symmetry

The axis of symmetry of a quadratic function in vertex form is given by the equation $x = h$. In the vertex form of the function $g(x) = 10(x-5)^2 - 37$, the value of $h$ is 5. Therefore, the axis of symmetry is the line $x = 5$, not $x = -5$. So, statement A is false.

Statement B: The Vertex of $g(x)$

The vertex of a quadratic function in vertex form is given by the point $(h,k)$. In the vertex form of the function $g(x) = 10(x-5)^2 - 37$, the value of $h$ is 5 and the value of $k$ is -37. Therefore, the vertex of $g(x)$ is the point $(5,-37)$. So, statement B is true.

Statement C: The Value of $g(5)$

To determine the value of $g(5)$, we can substitute $x = 5$ into the vertex form of the function $g(x) = 10(x-5)^2 - 37$.

# Substitute x = 5 into the vertex form of the function
g_5 = 10*(5 - 5)**2 - 37

Simplifying the expression, we get $g(5) = -37$. Therefore, statement C is true.

Conclusion

Introduction

In our previous article, we explored the properties of the function $g(x) = 10x^2 - 100x + 213$ when written in vertex form as $g(x) = 10(x-5)^2 - 37$. We examined three statements about $g(x)$ and determined which ones were true. In this article, we will answer some frequently asked questions about the function $g(x)$ and its properties.

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

A: The vertex form of a quadratic function is given by $g(x) = a(x-h)^2 + k$, 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. This involves factoring out the coefficient of the $x^2$ term, adding and subtracting the square of half the coefficient of the $x$ term inside the parentheses, and rewriting the expression inside the parentheses as a perfect square trinomial.

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

A: The axis of symmetry of a quadratic function in vertex form is given by the equation $x = h$, where $h$ is the value of the vertex form.

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

A: The vertex of a quadratic function in vertex form is given by the point $(h,k)$, where $h$ and $k$ are the values of the vertex form.

Q: How do I find the value of $g(x)$ at a given point?

A: To find the value of $g(x)$ at a given point, you need to substitute the value of $x$ into the vertex form of the function and simplify the expression.

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

A: The vertex form of a quadratic function is a powerful tool for analyzing and graphing quadratic equations. It allows us to easily identify the vertex of the parabola, the axis of symmetry, and the direction of the parabola.

Q: Can I use the vertex form of a quadratic function to solve systems of equations?

A: Yes, you can use the vertex form of a quadratic function to solve systems of equations. By substituting the value of $x$ into the vertex form of the function, you can solve for the value of $y$.

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

A: To graph a quadratic function in vertex form, you need to identify the vertex of the parabola, the axis of symmetry, and the direction of the parabola. You can then use this information to plot the graph of the function.

Conclusion

In conclusion, we have answered some frequently asked questions about the function $g(x)$ and its properties. We have discussed the vertex form of a quadratic function, how to convert a quadratic function to vertex form, and the significance of the vertex form of a quadratic function. We hope that this article has been helpful in understanding the properties of the function $g(x)$ and its applications.