Rewrite The Quadratic Function Below In Standard Form:${ Y = 2(x - 3)^2 + 5 }$ { Y = \square \}

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding how to rewrite them in standard form is crucial for solving equations and analyzing their properties. In this article, we will focus on rewriting the quadratic function $y = 2(x - 3)^2 + 5$ in standard form.

What is Standard Form?

Standard form of a quadratic function is a way of expressing the function in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. This form is useful for identifying the vertex, axis of symmetry, and other important features of the quadratic function.

Rewriting the Quadratic Function

To rewrite the quadratic function $y = 2(x - 3)^2 + 5$ in standard form, we need to expand the squared term and simplify the expression.

Expanding the Squared Term

The first step is to expand the squared term $(x - 3)^2$. Using the formula $(x - a)^2 = x^2 - 2ax + a^2$, we get:

$(x - 3)^2 = x^2 - 2(3)x + 3^2$ $(x - 3)^2 = x^2 - 6x + 9$

Simplifying the Expression

Now that we have expanded the squared term, we can simplify the expression by multiplying the coefficient of the squared term, which is 2, with the expanded term:

$y = 2(x^2 - 6x + 9) + 5$ $y = 2x^2 - 12x + 18 + 5$ $y = 2x^2 - 12x + 23$

Standard Form

The rewritten quadratic function in standard form is:

$y = 2x^2 - 12x + 23$

Key Features of the Quadratic Function

Now that we have rewritten the quadratic function in standard form, we can identify some of its key features.

Vertex

The vertex of a quadratic function in standard form is given by the formula $x = -\frac{b}{2a}$. In this case, $a = 2$ and $b = -12$, so:

$x = -\frac{-12}{2(2)}$ $x = 3$

The vertex is at the point $(3, 23)$.

Axis of Symmetry

The axis of symmetry of a quadratic function in standard form is given by the formula $x = -\frac{b}{2a}$. In this case, we already know that the axis of symmetry is at $x = 3$.

Domain and Range

The domain of a quadratic function is all real numbers, and the range is all real numbers greater than or equal to the minimum value of the function.

Conclusion

Rewriting a quadratic function in standard form is an important skill in mathematics. By following the steps outlined in this article, we can rewrite the quadratic function $y = 2(x - 3)^2 + 5$ in standard form as $y = 2x^2 - 12x + 23$. This form is useful for identifying the vertex, axis of symmetry, and other important features of the quadratic function.

Key Takeaways

• Standard form of a quadratic function is a way of expressing the function in the form $y = ax^2 + bx + c$.
• To rewrite a quadratic function in standard form, we need to expand the squared term and simplify the expression.
• The vertex of a quadratic function in standard form is given by the formula $x = -\frac{b}{2a}$.
• The axis of symmetry of a quadratic function in standard form is given by the formula $x = -\frac{b}{2a}$.
• The domain of a quadratic function is all real numbers, and the range is all real numbers greater than or equal to the minimum value of the function.

Further Reading

For more information on quadratic functions, including how to graph and analyze them, see the following resources:

• Quadratic Functions
• Graphing Quadratic Functions
• Analyzing Quadratic Functions
  Quadratic Function Q&A =========================

Introduction

Quadratic functions are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving equations and analyzing their properties. In this article, we will answer some common questions about quadratic functions.

Q: What is a quadratic function?

A quadratic function is a polynomial function of degree two, which means that the highest power of the variable (usually x) is two. It can be written in the form $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

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

The standard form of a quadratic function is $y = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. This form is useful for identifying the vertex, axis of symmetry, and other important features of the quadratic function.

Q: How do I rewrite a quadratic function in standard form?

To rewrite a quadratic function in standard form, you need to expand the squared term and simplify the expression. Here's an example:

$y = 2(x - 3)^2 + 5$

Expanding the squared term:

$(x - 3)^2 = x^2 - 2(3)x + 3^2$ $(x - 3)^2 = x^2 - 6x + 9$

Simplifying the expression:

$y = 2(x^2 - 6x + 9) + 5$ $y = 2x^2 - 12x + 18 + 5$ $y = 2x^2 - 12x + 23$

Q: What is the vertex of a quadratic function?

The vertex of a quadratic function is the point at which the function changes direction. It is given by the formula $x = -\frac{b}{2a}$.

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

The axis of symmetry of a quadratic function is given by the formula $x = -\frac{b}{2a}$. This is the vertical line that passes through the vertex of the function.

Q: What is the domain and range of a quadratic function?

The domain of a quadratic function is all real numbers, and the range is all real numbers greater than or equal to the minimum value of the function.

Q: How do I graph a quadratic function?

To graph a quadratic function, you can use the following steps:

1. Find the vertex of the function.
2. Find the axis of symmetry of the function.
3. Plot the vertex and the axis of symmetry on a coordinate plane.
4. Use the formula $y = ax^2 + bx + c$ to find the values of the function at various points on the graph.

Q: What are some common applications of quadratic functions?

Quadratic functions have many real-world applications, including:

• Modeling the trajectory of a projectile
• Finding the maximum or minimum value of a function
• Analyzing the motion of an object under the influence of gravity
• Solving optimization problems

Conclusion

Quadratic functions are a fundamental concept in mathematics, and understanding how to work with them is crucial for solving equations and analyzing their properties. By following the steps outlined in this article, you can answer common questions about quadratic functions and gain a deeper understanding of this important mathematical concept.

Key Takeaways

• A quadratic function is a polynomial function of degree two.
• The standard form of a quadratic function is $y = ax^2 + bx + c$.
• To rewrite a quadratic function in standard form, you need to expand the squared term and simplify the expression.
• The vertex of a quadratic function is given by the formula $x = -\frac{b}{2a}$.
• The axis of symmetry of a quadratic function is given by the formula $x = -\frac{b}{2a}$.
• The domain of a quadratic function is all real numbers, and the range is all real numbers greater than or equal to the minimum value of the function.

Further Reading

For more information on quadratic functions, including how to graph and analyze them, see the following resources:

• Quadratic Functions
• Graphing Quadratic Functions
• Analyzing Quadratic Functions