Christian Is Rewriting An Expression Of The Form Y = A X 2 + B X + C Y = Ax^2 + Bx + C Y = A X 2 + B X + C Into The Form Y = A ( X − H ) 2 + K Y = A(x - H)^2 + K Y = A ( X − H ) 2 + K . Which Of The Following Must Be True?A. The Value Of A A A Remains The Same.B. H H H Is Equal To

Introduction

In mathematics, quadratic equations are a fundamental concept that can be expressed in various forms. One of the most important forms is the vertex form, which is represented as $y = a(x - h)^2 + k$. This form provides valuable information about the quadratic function, including its vertex, axis of symmetry, and direction of opening. In this article, we will explore the process of converting a quadratic equation from the standard form $y = ax^2 + bx + c$ to the vertex form, and discuss the properties that must be true in this conversion.

Understanding the Vertex Form

The vertex form of a quadratic equation is given by $y = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex. The value of $a$ determines the direction and width of the parabola, while the values of $h$ and $k$ determine the location of the vertex. The axis of symmetry is given by the equation $x = h$, which is a vertical line that passes through the vertex.

Converting to Vertex Form

To convert a quadratic equation from the standard form to the vertex form, we need to complete the square. This involves rewriting the equation in a way that allows us to extract the values of $h$ and $k$. The process involves the following steps:

1. Factor out the coefficient of $x^2$: If the coefficient of $x^2$ is not 1, we need to factor it out. This will give us the value of $a$ in the vertex form.
2. Complete the square: We need to add and subtract a constant term to create a perfect square trinomial. This constant term is given by $\left(\frac{b}{2}\right)^2$.
3. Write the equation in vertex form: Once we have completed the square, we can write the equation in the vertex form $y = a(x - h)^2 + k$.

Properties of the Vertex Form

Now that we have converted the quadratic equation to the vertex form, we can discuss the properties that must be true. The following properties are essential:

• The value of $a$ remains the same: When we convert a quadratic equation from the standard form to the vertex form, the value of $a$ remains the same. This is because the coefficient of $x^2$ is factored out in the first step, and the value of $a$ is preserved throughout the conversion process.
• The value of $h$ is equal to $-\frac{b}{2a}$: The value of $h$ is given by the formula $h = -\frac{b}{2a}$. This is because the constant term that we added and subtracted in the second step is equal to $\left(\frac{b}{2}\right)^2$, which is the square of half the coefficient of $x$.
• The value of $k$ is equal to $c - \frac{b^2}{4a}$: The value of $k$ is given by the formula $k = c - \frac{b^2}{4a}$. This is because the constant term that we added and subtracted in the second step is equal to $\left(\frac{b}{2}\right)^2$, which is the square of half the coefficient of $x$.

Example

Let's consider the quadratic equation $y = 2x^2 + 4x + 3$. We can convert this equation to the vertex form by completing the square.

1. Factor out the coefficient of $x^2$: We have $y = 2(x^2 + 2x) + 3$.
2. Complete the square: We add and subtract $\left(\frac{2}{2}\right)^2 = 1$ to create a perfect square trinomial. This gives us $y = 2(x^2 + 2x + 1) + 3 - 2$.
3. Write the equation in vertex form: We can now write the equation in the vertex form $y = 2(x + 1)^2 + 1$.

In this example, we can see that the value of $a$ remains the same, which is 2. The value of $h$ is equal to $-\frac{b}{2a} = -\frac{4}{2(2)} = -1$, and the value of $k$ is equal to $c - \frac{b^2}{4a} = 3 - \frac{4^2}{4(2)} = 1$.

Conclusion

In conclusion, converting a quadratic equation from the standard form to the vertex form involves completing the square and writing the equation in the form $y = a(x - h)^2 + k$. The properties that must be true in this conversion are:

• The value of $a$ remains the same: The value of $a$ is preserved throughout the conversion process.
• The value of $h$ is equal to $-\frac{b}{2a}$: The value of $h$ is given by the formula $h = -\frac{b}{2a}$.
• The value of $k$ is equal to $c - \frac{b^2}{4a}$: The value of $k$ is given by the formula $k = c - \frac{b^2}{4a}$.

Q&A: Converting Quadratic Equations to Vertex Form

In the previous article, we discussed the process of converting a quadratic equation from the standard form to the vertex form. We also explored the properties that must be true in this conversion. In this article, we will answer some frequently asked questions about converting quadratic equations to vertex form.

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

A: The vertex form of a quadratic equation is given by $y = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex.

Q: How do I convert a quadratic equation from the standard form to the vertex form?

A: To convert a quadratic equation from the standard form to the vertex form, you need to complete the square. This involves rewriting the equation in a way that allows you to extract the values of $h$ and $k$. The process involves the following steps:

1. Factor out the coefficient of $x^2$: If the coefficient of $x^2$ is not 1, you need to factor it out. This will give you the value of $a$ in the vertex form.
2. Complete the square: You need to add and subtract a constant term to create a perfect square trinomial. This constant term is given by $\left(\frac{b}{2}\right)^2$.
3. Write the equation in vertex form: Once you have completed the square, you can write the equation in the vertex form $y = a(x - h)^2 + k$.

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

A: The vertex form of a quadratic equation provides valuable information about the quadratic function, including its vertex, axis of symmetry, and direction of opening. This form is particularly useful in graphing and analyzing quadratic functions.

Q: How do I find the value of $h$ in the vertex form of a quadratic equation?

A: The value of $h$ is given by the formula $h = -\frac{b}{2a}$. This is because the constant term that you added and subtracted in the second step is equal to $\left(\frac{b}{2}\right)^2$, which is the square of half the coefficient of $x$.

Q: How do I find the value of $k$ in the vertex form of a quadratic equation?

A: The value of $k$ is given by the formula $k = c - \frac{b^2}{4a}$. This is because the constant term that you added and subtracted in the second step is equal to $\left(\frac{b}{2}\right)^2$, which is the square of half the coefficient of $x$.

Q: Can I convert a quadratic equation from the vertex form to the standard form?

A: Yes, you can convert a quadratic equation from the vertex form to the standard form by expanding the squared term and simplifying the equation.

Q: What are some common mistakes to avoid when converting quadratic equations to vertex form?

A: Some common mistakes to avoid when converting quadratic equations to vertex form include:

• Not factoring out the coefficient of $x^2$: Make sure to factor out the coefficient of $x^2$ in the first step.
• Not completing the square correctly: Make sure to add and subtract the correct constant term to create a perfect square trinomial.
• Not writing the equation in vertex form: Make sure to write the equation in the vertex form $y = a(x - h)^2 + k$.

Conclusion

In conclusion, converting quadratic equations to vertex form is an essential skill in algebra and mathematics. By understanding the properties of the vertex form and following the steps outlined in this article, you can convert quadratic equations from the standard form to the vertex form and gain valuable insights into the properties of the quadratic function.