Rewrite The Equation In Standard Form: $ Y = 3(x-1)^2 - 3 $

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Introduction

In mathematics, equations are a fundamental concept that helps us describe relationships between variables. One of the essential skills in algebra is rewriting equations in standard form, which is a crucial step in solving equations and understanding their properties. In this article, we will focus on rewriting the equation $ y = 3(x-1)^2 - 3 $ in standard form.

Understanding the Equation

Before we dive into rewriting the equation, let's first understand what it means. The given equation is a quadratic equation in the form of $ y = a(x-h)^2 + k $, where $ a $ is the coefficient of the squared term, $ h $ is the x-coordinate of the vertex, and $ k $ is the y-coordinate of the vertex. In this case, $ a = 3 $, $ h = 1 $, and $ k = -3 $.

Rewriting the Equation in Standard Form

To rewrite the equation in standard form, we need to expand the squared term and simplify the expression. Let's start by expanding the squared term:

y=3(x−1)2−3y = 3(x-1)^2 - 3

y=3(x2−2x+1)−3y = 3(x^2 - 2x + 1) - 3

Now, let's simplify the expression by distributing the coefficient $ 3 $ to each term inside the parentheses:

y=3x2−6x+3−3y = 3x^2 - 6x + 3 - 3

y=3x2−6xy = 3x^2 - 6x

The Final Answer

After simplifying the expression, we get the equation in standard form:

y=3x2−6xy = 3x^2 - 6x

This is the final answer, but let's take a closer look at the equation and understand its properties.

Properties of the Equation

The equation $ y = 3x^2 - 6x $ is a quadratic equation in the form of $ y = ax^2 + bx + c $, where $ a = 3 $, $ b = -6 $, and $ c = 0 $. This equation represents a parabola that opens upward, since the coefficient of the squared term $ a $ is positive.

Graphing the Equation

To graph the equation, we can use the vertex form of the equation, which is $ y = a(x-h)^2 + k $. In this case, the vertex is at the point $ (h, k) = (1, -3) $. We can plot this point on the coordinate plane and use it as a reference to graph the parabola.

Conclusion

Rewriting the equation $ y = 3(x-1)^2 - 3 $ in standard form is a crucial step in understanding the properties of the equation and solving it. By expanding the squared term and simplifying the expression, we get the equation in standard form, which is $ y = 3x^2 - 6x $. This equation represents a parabola that opens upward, and its vertex is at the point $ (1, -3) $. By graphing the equation, we can visualize its properties and understand its behavior.

Common Mistakes to Avoid

When rewriting the equation in standard form, there are several common mistakes to avoid. Here are a few:

  • Not distributing the coefficient: When expanding the squared term, make sure to distribute the coefficient to each term inside the parentheses.
  • Not simplifying the expression: After expanding the squared term, make sure to simplify the expression by combining like terms.
  • Not checking the vertex: When graphing the equation, make sure to check the vertex and use it as a reference to graph the parabola.

Real-World Applications

The equation $ y = 3(x-1)^2 - 3 $ has several real-world applications. Here are a few:

  • Physics: The equation can be used to model the motion of an object under the influence of gravity.
  • Engineering: The equation can be used to design and optimize systems, such as bridges and buildings.
  • Economics: The equation can be used to model the behavior of economic systems, such as supply and demand.

Final Thoughts

Q&A: Frequently Asked Questions

Q: What is the standard form of a quadratic equation? A: The standard form of a quadratic equation is $ y = ax^2 + bx + c $, where $ a $, $ b $, and $ c $ are constants.

Q: How do I rewrite a quadratic equation in standard form? A: To rewrite a quadratic equation in standard form, you need to expand the squared term and simplify the expression. Here's a step-by-step guide:

  1. Expand the squared term by distributing the coefficient to each term inside the parentheses.
  2. Simplify the expression by combining like terms.
  3. Write the equation in the form $ y = ax^2 + bx + c $.

Q: What is the vertex form of a quadratic equation? A: The vertex form of a quadratic equation is $ y = a(x-h)^2 + k $, where $ (h, k) $ is the vertex of the parabola.

Q: How do I graph a quadratic equation? A: To graph a quadratic equation, you can use the vertex form of the equation. Here's a step-by-step guide:

  1. Plot the vertex of the parabola at the point $ (h, k) $.
  2. Use the vertex as a reference to graph the parabola.
  3. Make sure to include the x-intercepts and the y-intercept on the graph.

Q: What are some common mistakes to avoid when rewriting a quadratic equation in standard form? A: Here are some common mistakes to avoid:

  • Not distributing the coefficient when expanding the squared term.
  • Not simplifying the expression by combining like terms.
  • Not checking the vertex when graphing the equation.

Q: What are some real-world applications of quadratic equations? A: Quadratic equations have several real-world applications, including:

  • Physics: Quadratic equations can be used to model the motion of an object under the influence of gravity.
  • Engineering: Quadratic equations can be used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations can be used to model the behavior of economic systems, such as supply and demand.

Q: How do I solve a quadratic equation? A: To solve a quadratic equation, you can use the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula will give you two solutions for the equation.

Q: What is the quadratic formula? A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula will give you two solutions for the equation.

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 follow these steps:

  1. Write the equation in the form $ ax^2 + bx + c = 0 $.
  2. Plug in the values of $ a $, $ b $, and $ c $ into the quadratic formula.
  3. Simplify the expression and solve for $ x $.

Conclusion

Rewriting the equation $ y = 3(x-1)^2 - 3 $ in standard form is a crucial step in understanding the properties of the equation and solving it. By expanding the squared term and simplifying the expression, we get the equation in standard form, which is $ y = 3x^2 - 6x $. This equation represents a parabola that opens upward, and its vertex is at the point $ (1, -3) $. By graphing the equation, we can visualize its properties and understand its behavior.