Write \[$ H(x) = 7 + 10x + X^2 \$\] In Vertex Form.1. Write \[$ H \$\] In Standard Form: $\[ H(x) = X^2 + 10x + 7 \\]2. Form A Perfect Square Trinomial By Adding And Subtracting \[$ \left(\frac{b}{2}\right)^2 \$\]:
Understanding the Problem
In this article, we will explore the process of converting a quadratic function from standard form to vertex form. The standard form of a quadratic function is given by { h(x) = ax^2 + bx + c $}$, where a, b, and c are constants. The vertex form of a quadratic function is given by { h(x) = a(x - h)^2 + k $}$, where (h, k) is the vertex of the parabola.
Step 1: Write the Quadratic Function in Standard Form
The given quadratic function is { h(x) = 7 + 10x + x^2 $}$. To write this function in standard form, we need to rearrange the terms in descending order of the exponent of x.
{ h(x) = x^2 + 10x + 7 $}$
Step 2: Identify the Values of a, b, and c
In the standard form of the quadratic function, we can identify the values of a, b, and c as follows:
- a = 1
- b = 10
- c = 7
Step 3: Form a Perfect Square Trinomial
To form a perfect square trinomial, we need to add and subtract { \left(\frac{b}{2}\right)^2 $}$ inside the parentheses.
{ \left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = 5^2 = 25 $}$
Now, we can add and subtract 25 inside the parentheses:
{ h(x) = x^2 + 10x + 25 - 25 + 7 $}$
Step 4: Simplify the Expression
We can simplify the expression by combining like terms:
{ h(x) = (x^2 + 10x + 25) - 25 + 7 $}$
{ h(x) = (x + 5)^2 - 18 $}$
Step 5: Write the Quadratic Function in Vertex Form
The vertex form of the quadratic function is given by { h(x) = a(x - h)^2 + k $}$. We can identify the values of a, h, and k as follows:
- a = 1
- h = -5
- k = -18
Therefore, the vertex form of the quadratic function is:
{ h(x) = (x + 5)^2 - 18 $}$
Conclusion
In this article, we have explored the process of converting a quadratic function from standard form to vertex form. We have identified the values of a, b, and c in the standard form, formed a perfect square trinomial, simplified the expression, and written the quadratic function in vertex form. The vertex form of the quadratic function is given by { h(x) = (x + 5)^2 - 18 $}$, where (h, k) = (-5, -18) is the vertex of the parabola.
Example Problems
Problem 1
Write the quadratic function { h(x) = 3x^2 + 12x + 15 $}$ in vertex form.
Solution
To write the quadratic function in vertex form, we need to follow the same steps as before.
- Identify the values of a, b, and c: a = 3, b = 12, c = 15
- Form a perfect square trinomial: { \left(\frac{b}{2}\right)^2 = \left(\frac{12}{2}\right)^2 = 6^2 = 36 $}$
- Simplify the expression: { h(x) = (x^2 + 12x + 36) - 36 + 15 $}$
- Write the quadratic function in vertex form: { h(x) = (x + 6)^2 - 21 $}$
Problem 2
Write the quadratic function { h(x) = 2x^2 - 8x + 10 $}$ in vertex form.
Solution
To write the quadratic function in vertex form, we need to follow the same steps as before.
- Identify the values of a, b, and c: a = 2, b = -8, c = 10
- Form a perfect square trinomial: { \left(\frac{b}{2}\right)^2 = \left(\frac{-8}{2}\right)^2 = (-4)^2 = 16 $}$
- Simplify the expression: { h(x) = (x^2 - 8x + 16) - 16 + 10 $}$
- Write the quadratic function in vertex form: { h(x) = (x - 4)^2 - 6 $}$
Final Answer
Q: What is the standard form of a quadratic function?
A: The standard form of a quadratic function is given by { h(x) = ax^2 + bx + c $}$, where a, b, and c are constants.
Q: What is the vertex form of a quadratic function?
A: The vertex form of a quadratic function is given by { h(x) = a(x - h)^2 + k $}$, where (h, k) is the vertex of the parabola.
Q: How do I convert a quadratic function from standard form to vertex form?
A: To convert a quadratic function from standard form to vertex form, you need to follow these steps:
- Identify the values of a, b, and c in the standard form.
- Form a perfect square trinomial by adding and subtracting { \left(\frac{b}{2}\right)^2 $}$.
- Simplify the expression by combining like terms.
- Write the quadratic function in vertex form.
Q: What is a perfect square trinomial?
A: A perfect square trinomial is an expression of the form { (x + p)^2 $}$ or { (x - p)^2 $}$, where p is a constant.
Q: How do I form a perfect square trinomial?
A: To form a perfect square trinomial, you need to add and subtract { \left(\frac{b}{2}\right)^2 $}$ inside the parentheses.
Q: What is the significance of the vertex form of a quadratic function?
A: The vertex form of a quadratic function is useful in graphing and analyzing the behavior of the function. It helps to identify the vertex of the parabola, which is the maximum or minimum point of the function.
Q: Can I use the vertex form of a quadratic function to find the x-intercepts?
A: Yes, you can use the vertex form of a quadratic function to find the x-intercepts. To do this, you need to set the function equal to zero and solve for x.
Q: How do I find the x-intercepts of a quadratic function in vertex form?
A: To find the x-intercepts of a quadratic function in vertex form, you need to set the function equal to zero and solve for x. This can be done by using the quadratic formula or by factoring the expression.
Q: Can I use the vertex form of a quadratic function to find the y-intercept?
A: Yes, you can use the vertex form of a quadratic function to find the y-intercept. To do this, you need to substitute x = 0 into the function and solve for y.
Q: How do I find the y-intercept of a quadratic function in vertex form?
A: To find the y-intercept of a quadratic function in vertex form, you need to substitute x = 0 into the function and solve for y. This can be done by using the quadratic formula or by factoring the expression.
Q: What are some common mistakes to avoid when converting a quadratic function from standard form to vertex form?
A: Some common mistakes to avoid when converting a quadratic function from standard form to vertex form include:
- Not identifying the values of a, b, and c correctly.
- Not forming a perfect square trinomial correctly.
- Not simplifying the expression correctly.
- Not writing the quadratic function in vertex form correctly.
Q: How can I practice converting quadratic functions from standard form to vertex form?
A: You can practice converting quadratic functions from standard form to vertex form by using online resources, such as worksheets and practice problems. You can also try converting different quadratic functions to vertex form to get a feel for the process.
Q: What are some real-world applications of converting quadratic functions from standard form to vertex form?
A: Some real-world applications of converting quadratic functions from standard form to vertex form include:
- Modeling the trajectory of a projectile.
- Analyzing the behavior of a population growth model.
- Finding the maximum or minimum value of a function.
Conclusion
In this article, we have discussed the process of converting a quadratic function from standard form to vertex form. We have also answered some frequently asked questions about this process. By following the steps outlined in this article, you can convert a quadratic function from standard form to vertex form and gain a deeper understanding of the behavior of the function.