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

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 vertex of the parabola. This form is particularly useful for identifying the vertex and the direction of the parabola. In this article, we will explore how to rewrite the quadratic equation $y = 2x^2 - 8x + 9$ in vertex form.

The Process of Rewriting in Vertex Form

To rewrite a quadratic equation in vertex form, we need to complete the square. This involves manipulating the equation to create a perfect square trinomial. The general process is as follows:

1. Factor out the coefficient of the $x^2$ term.
2. Move the constant term to the right-hand side of the equation.
3. Add and subtract the square of half the coefficient of the $x$ term to the left-hand side.
4. Factor the perfect square trinomial.

Applying the Process to the Given Equation

Let's apply the process to the given equation $y = 2x^2 - 8x + 9$.

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

The coefficient of the $x^2$ term is 2. We can factor this out:

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

Step 2: Move the constant term to the right-hand side of the equation

We can move the constant term to the right-hand side:

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

Step 3: Add and subtract the square of half the coefficient of the $x$ term to the left-hand side

The coefficient of the $x$ term is -4. Half of this is -2, and the square of -2 is 4. We can add and subtract 4 to the left-hand side:

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

Step 4: Factor the perfect square trinomial

We can factor the perfect square trinomial:

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

Step 5: Simplify the equation

We can simplify the equation by distributing the 2:

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

Step 6: Add 9 to both sides of the equation

We can add 9 to both sides of the equation to isolate y:

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

Conclusion

The quadratic equation $y = 2x^2 - 8x + 9$ rewritten in vertex form is $y = 2(x - 2)^2 + 1$. This form is useful for identifying the vertex and the direction of the parabola.

Answer

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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 vertex of the parabola. This form is particularly useful for identifying the vertex and the direction of the parabola.

Q: How do I rewrite a quadratic equation in vertex form?

A: To rewrite a quadratic equation in vertex form, you need to complete the square. This involves manipulating the equation to create a perfect square trinomial. The general process is as follows:

1. Factor out the coefficient of the $x^2$ term.
2. Move the constant term to the right-hand side of the equation.
3. Add and subtract the square of half the coefficient of the $x$ term to the left-hand side.
4. Factor the perfect square trinomial.

Q: What is the coefficient of the $x^2$ term, and why is it important?

A: The coefficient of the $x^2$ term is the number that multiplies the $x^2$ term. It is important because it determines the direction of the parabola. If the coefficient is positive, the parabola opens upwards. If the coefficient is negative, the parabola opens downwards.

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

A: To find the vertex of a parabola in vertex form, you can simply read off the values of $h$ and $k$. The vertex is the point $(h,k)$.

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

A: The vertex of a parabola is the highest or lowest point on the graph. It is the point where the parabola changes direction.

Q: Can I use vertex form to graph a parabola?

A: Yes, you can use vertex form to graph a parabola. To do this, you need to plot the vertex and then use the direction of the parabola to determine the shape of the graph.

Q: Are there any other ways to rewrite a quadratic equation?

A: Yes, there are other ways to rewrite a quadratic equation. One common method is to use the factoring method, which involves factoring the quadratic expression into the product of two binomials.

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

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

• Not factoring out the coefficient of the $x^2$ term.
• Not moving the constant term to the right-hand side of the equation.
• Not adding and subtracting the square of half the coefficient of the $x$ term to the left-hand side.
• Not factoring the perfect square trinomial.

Q: How do I check my work when rewriting a quadratic equation in vertex form?

A: To check your work, you can plug in a test value for $x$ and see if the resulting value of $y$ matches the original equation. You can also use a graphing calculator to graph the original equation and the rewritten equation in vertex form and compare the two graphs.

Conclusion

Rewriting a quadratic equation in vertex form can be a useful tool for identifying the vertex and the direction of the parabola. By following the steps outlined in this article, you can rewrite a quadratic equation in vertex form and gain a deeper understanding of the properties of quadratic equations.