Vertex Form: Convert $h(x)=7+10x+x^2$

Hey guys! Today, we're diving into a super useful skill in algebra: converting a quadratic equation into vertex form. Specifically, we're going to take the equation $h(x) = 7 + 10x + x^2$ and transform it. Vertex form is awesome because it immediately tells you the vertex (the minimum or maximum point) of the parabola represented by the quadratic equation. Let's get started!

Understanding Vertex Form

First off, let's clarify what vertex form actually looks like. A quadratic equation in vertex form is written as:

$h(x) = a(x - h)^2 + k$

Where:

• $(h, k)$ is the vertex of the parabola.
• $a$ determines the direction and stretch of the parabola. If $a > 0$, the parabola opens upwards, and if $a < 0$, it opens downwards. The absolute value of $a$ tells us how stretched or compressed the parabola is compared to the standard parabola $y = x^2$.

Knowing this form allows us to quickly identify key features of the quadratic function's graph, making it a valuable tool in analysis and graphing.

Completing the Square: The Key Technique

The main technique we'll use to convert our equation into vertex form is called completing the square. This method involves manipulating the quadratic expression to create a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form $(x + b)^2$ or $(x - b)^2$. Let’s walk through the steps:

1. Rewrite the Equation: Start by rewriting the given equation in the standard form of a quadratic equation, which is $ax^2 + bx + c$. So, $h(x) = 7 + 10x + x^2$ becomes $h(x) = x^2 + 10x + 7$.

2. Focus on the $x^2 + bx$ terms: We're going to focus on the $x^2 + 10x$ part of the equation. Our goal is to turn this into a perfect square trinomial.

3. Complete the Square: To complete the square, we need to add and subtract a value that will make $x^2 + 10x$ a perfect square. The value we need to add is $(\frac{b}{2})^2$, where $b$ is the coefficient of the $x$ term. In our case, $b = 10$, so we need to add and subtract $(\frac{10}{2})^2 = 5^2 = 25$.

   So, we rewrite the equation as:

   $h(x) = x^2 + 10x + 25 - 25 + 7$

   Notice that we're adding and subtracting the same value, so we're not changing the equation.

4. Factor the Perfect Square Trinomial: Now, the $x^2 + 10x + 25$ part is a perfect square trinomial, which can be factored as $(x + 5)^2$. So, our equation becomes:

   $h(x) = (x + 5)^2 - 25 + 7$

5. Simplify: Finally, simplify the equation by combining the constant terms:

   $h(x) = (x + 5)^2 - 18$

And there you have it! The equation $h(x) = 7 + 10x + x^2$ in vertex form is $h(x) = (x + 5)^2 - 18$.

Identifying the Vertex

Now that we have the equation in vertex form, it's easy to identify the vertex of the parabola. Remember that the vertex form is $h(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex. In our case, $h(x) = (x + 5)^2 - 18$, which can be rewritten as $h(x) = (x - (-5))^2 + (-18)$. Therefore, the vertex is $(-5, -18)$.

This means the parabola has its minimum point at $x = -5$ and $y = -18$. Knowing the vertex is super helpful for graphing the parabola and understanding its behavior.

Why Vertex Form Matters?

You might be wondering, why bother converting to vertex form? Here are a few reasons:

• Easy Vertex Identification: As we've seen, the vertex form directly gives you the coordinates of the vertex, which is crucial for understanding the parabola's graph.
• Graphing Made Simple: With the vertex and the direction of the parabola (determined by the sign of $a$), you can quickly sketch the graph of the quadratic function.
• Optimization Problems: In many real-world problems, you need to find the maximum or minimum value of a quadratic function. The vertex form immediately tells you this value.
• Transformations: Vertex form makes it easy to see how the parabola has been transformed from the basic parabola $y = x^2$. The values of $h$ and $k$ represent horizontal and vertical shifts, respectively.

Common Mistakes to Avoid

When completing the square and converting to vertex form, it's easy to make a few common mistakes. Here are some to watch out for:

• Forgetting to Add and Subtract: Remember that when you add a value to complete the square, you must also subtract it to keep the equation balanced. Failing to do so will change the equation and lead to an incorrect vertex form.
• Incorrectly Calculating $(\frac{b}{2})^2$: Double-check your calculation of $(\frac{b}{2})^2$. This is a crucial step, and an error here will throw off the entire process.
• Sign Errors: Pay close attention to signs when factoring and simplifying. A simple sign error can lead to an incorrect vertex.
• Forgetting to Factor Correctly: Make sure you factor the perfect square trinomial correctly. Remember that $(x + b)^2 = x^2 + 2bx + b^2$.

Practice Problems

To really master converting to vertex form, it's essential to practice. Here are a couple of problems you can try:

1. Convert $f(x) = x^2 - 6x + 11$ to vertex form.
2. Convert $g(x) = 2x^2 + 8x + 5$ to vertex form. (Hint: Factor out the coefficient of $x^2$ first).

Work through these problems, paying close attention to each step, and you'll be a pro in no time!

Conclusion

So, converting the quadratic equation $h(x) = 7 + 10x + x^2$ into vertex form gives us $h(x) = (x + 5)^2 - 18$. This form makes it easy to identify the vertex of the parabola, which is $(-5, -18)$. Mastering this technique is super useful for graphing quadratic functions and solving optimization problems. Keep practicing, and you'll become a vertex form wizard! Keep rocking!