Rewrite the function in f(x)=a(x-h)^2+k form.
Graph it by using the transformation.
F(x)=x^2+8x+15
F(x)=x^2+8x+15
complete the square, you must know this procedure!
= (x^2 + 8x + 16 - 16) + 15
= ( (x+4)^2 - 16) + 15
= (x+4)^2 - 1
To rewrite the function in f(x) = a(x-h)^2 + k form, we need to complete the square. Here's how you can do it step by step:
1. Start with the given function: F(x) = x^2 + 8x + 15.
2. Group the first two terms and write it as: (x^2 + 8x) + 15.
3. To complete the square, take half of the coefficient of x (which is 8) and square it: (8/2)^2 = 16.
4. Add and subtract 16 inside the parentheses: (x^2 + 8x + 16 - 16) + 15.
5. Group the first three terms and simplify: (x^2 + 8x + 16) - 16 + 15.
6. Rearrange the terms: (x^2 + 8x + 16) + (-16 + 15).
7. Now, the expression inside the parentheses is a perfect square trinomial, which can be factored as (x + 4)^2.
8. Simplify the remaining terms: (x + 4)^2 - 1.
So, the function F(x) can be rewritten in the form f(x) = a(x - h)^2 + k as f(x) = (x + 4)^2 - 1.
To graph the function using transformations, we can observe:
1. The function f(x) = (x + 4)^2 represents a parabola that opens upwards since the coefficient of the x^2 term is positive.
2. The vertex of the parabola is located at the point (-4, -1) since (h, k) represents the coordinates of the vertex in the vertex form of a quadratic equation.
3. Since the constant term (-1) is subtracted from the square, the parabola is shifted one unit downward from the vertex.
4. By comparing it to the standard form of a parabola, f(x) = ax^2, we can determine that the coefficient a = 1, indicating that the graph remains unchanged in terms of width.
5. Based on the vertex and the shape of the parabola, we can plot points and draw the graph accordingly.
By following these steps, you should be able to rewrite the function and graph it using the transformation.