Which Equation Is Y = 2 X 2 − 8 X + 9 Y = 2x^2 - 8x + 9 Y = 2 X 2 − 8 X + 9 Rewritten In Vertex Form?A. Y = 2 ( X − 2 ) 2 + 9 Y = 2(x - 2)^2 + 9 Y = 2 ( X − 2 ) 2 + 9 B. Y = 2 ( X − 2 ) 2 + 5 Y = 2(x - 2)^2 + 5 Y = 2 ( X − 2 ) 2 + 5 C. Y = 2 ( X − 2 ) 2 + 1 Y = 2(x - 2)^2 + 1 Y = 2 ( X − 2 ) 2 + 1 D. Y = 2 ( X − 2 ) 2 + 17 Y = 2(x - 2)^2 + 17 Y = 2 ( X − 2 ) 2 + 17

Understanding Vertex Form

Vertex form is a way of expressing quadratic equations in the form of $y = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex of the parabola. This form is particularly useful for identifying the vertex of a parabola and for graphing quadratic functions.

Converting Standard Form to Vertex Form

To convert a quadratic equation from standard form to vertex form, we need to complete the square. The standard form of a quadratic equation is $y = ax^2 + bx + c$. To convert it to vertex form, we follow these steps:

1. Factor out the coefficient of $x^2$ from the first two terms.
2. Take half of the coefficient of $x$ and square it.
3. Add and subtract the squared value from the expression.
4. Rewrite the expression in the form of $y = a(x - h)^2 + k$.

Converting the Given Equation

Let's apply the steps to convert the given equation $y = 2x^2 - 8x + 9$ to vertex form.

Step 1: Factor out the coefficient of $x^2$

The coefficient of $x^2$ is 2. We factor it out from the first two terms:

$$y = 2(x^2 - 4x) + 9$$

Step 2: Take half of the coefficient of $x$ and square it

The coefficient of $x$ is -4. Half of -4 is -2, and squaring it gives 4.

Step 3: Add and subtract the squared value from the expression

We add and subtract 4 inside the parentheses:

$$y = 2(x^2 - 4x + 4 - 4) + 9$$

Step 4: Rewrite the expression in the form of $y = a(x - h)^2 + k$

Now we can rewrite the expression in the form of $y = a(x - h)^2 + k$:

$$y = 2(x - 2)^2 - 8 + 9$$

$$y = 2(x - 2)^2 + 1$$

Comparing with the Options

Now that we have rewritten the equation in vertex form, let's compare it with the given options:

• A. $y = 2(x - 2)^2 + 9$
• B. $y = 2(x - 2)^2 + 5$
• C. $y = 2(x - 2)^2 + 1$
• D. $y = 2(x - 2)^2 + 17$

The correct option is C. $y = 2(x - 2)^2 + 1$.

Conclusion

Q: What is vertex form, and why is it useful?

A: Vertex form is a way of expressing quadratic equations in the form of $y = a(x - h)^2 + k$, where $(h, k)$ represents the coordinates of the vertex of the parabola. This form is particularly useful for identifying the vertex of a parabola and for graphing quadratic functions.

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

A: To convert a quadratic equation from standard form to vertex form, you need to complete the square. The steps are:

1. Factor out the coefficient of $x^2$ from the first two terms.
2. Take half of the coefficient of $x$ and square it.
3. Add and subtract the squared value from the expression.
4. Rewrite the expression in the form of $y = a(x - h)^2 + k$.

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

A: The vertex of a parabola represents the minimum or maximum point of the parabola. In vertex form, the vertex is represented by the point $(h, k)$.

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

A: To find the vertex of a parabola in vertex form, simply look at the expression $y = a(x - h)^2 + k$. The vertex is represented by the point $(h, k)$.

Q: Can I use vertex form to graph a quadratic function?

A: Yes, vertex form is particularly useful for graphing quadratic functions. By identifying the vertex and the direction of the parabola, you can easily graph the function.

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

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

• Not factoring out the coefficient of $x^2$ correctly.
• Not taking half of the coefficient of $x$ correctly.
• Not adding and subtracting the squared value correctly.
• Not rewriting the expression in the correct form.

Q: Can I use vertex form to solve quadratic equations?

A: Yes, vertex form can be used to solve quadratic equations. By setting the expression equal to zero and solving for $x$, you can find the solutions to the equation.

Q: What are some real-world applications of vertex form?

A: Vertex form has many real-world applications, including:

• Modeling population growth and decline.
• Modeling the motion of objects under the influence of gravity.
• Modeling the spread of diseases.
• Modeling the behavior of electrical circuits.

Conclusion

In this article, we answered some frequently asked questions about rewriting quadratic equations in vertex form. We covered topics such as the significance of vertex form, how to convert a quadratic equation to vertex form, and some common mistakes to avoid. We also discussed the real-world applications of vertex form.