Complete the square and find the vertex form of the quadratic function.
f(x)= x^2 -2x+5
To complete the square, we need to take the coefficient of the x term, divide it by 2, and then square the result. For the quadratic function f(x) = x^2 - 2x + 5:
1. Take half of the coefficient of the x term: -2/2 = -1
2. Square the result: (-1)^2 = 1
Now let's rewrite the quadratic function by adding and subtracting 1 inside the parentheses:
f(x) = x^2 - 2x + 1 - 1 + 5
Next, let's group the square term with the constant term:
f(x) = (x^2 - 2x + 1) - 1 + 5
The grouped terms x^2 - 2x + 1 can be written as a square of a binomial:
f(x) = (x - 1)^2 - 1 + 5
Simplifying further:
f(x) = (x - 1)^2 + 4
Therefore, the vertex form of the quadratic function is f(x) = (x - 1)^2 + 4. The vertex of the parabola is (1, 4).
To complete the square and find the vertex form of the quadratic function, follow these steps:
Step 1: Rewrite the function in the form f(x) = a(x - h)^2 + k, where (h, k) represents the vertex.
In this case, f(x) = x^2 - 2x + 5.
Step 2: Group the terms inside the parentheses.
f(x) = (x^2 - 2x) + 5
Step 3: Take half of the coefficient of the x-term (2) and square it.
(2/2)^2 = 1^2 = 1
Step 4: Add the value obtained in step 3 to the expression inside the parentheses. Also, subtract the same value outside the parentheses to maintain the equality.
f(x) = (x^2 - 2x + 1 - 1) + 5
Step 5: Rearrange the expression inside the parentheses as a perfect square.
f(x) = (x^2 - 2x + 1) - 1 + 5
Step 6: Simplify the expression inside the parentheses.
f(x) = (x - 1)^2 + 4
Hence, the quadratic function f(x) = x^2 - 2x + 5 can be written in vertex form as f(x) = (x - 1)^2 + 4. In this form, the vertex is represented by (h, k), which is (1, 4).
To complete the square and find the vertex form of the quadratic function, follow these steps:
1. Start with the quadratic function in standard form: f(x) = ax^2 + bx + c.
In this case, f(x) = x^2 - 2x + 5.
2. Take the coefficient of the x-term (b) and divide it by 2, and then square the result. Add this squared value to the function.
b/2 = -2/2 = -1
(-1)^2 = 1
f(x) + 1 = x^2 - 2x + 1 + 5
3. Rearrange the terms in the parentheses to create a perfect square trinomial.
f(x) + 1 = (x^2 - 2x + 1) + 5
4. Simplify the expression inside the parentheses and combine like terms outside.
f(x) + 1 = (x-1)^2 + 5
5. Move the constant term outside the parentheses to obtain the vertex form.
f(x) = (x-1)^2 + 5 - 1
f(x) = (x-1)^2 + 4
Therefore, the vertex form of the quadratic function f(x) = x^2 -2x+5 is f(x) = (x-1)^2 + 4, and the vertex of the parabola is (1, 4).