a. Express f(x)=x^2 - 6x + 14 in the form f(x)=(x-h)^2 + k, where h and k are to be determined.
b.Hence, or otherwise, write down the coordinates of the vertex of the parabola equation y=x^2 - 6x + 14.
a. Alright, let's find the vertex form of the quadratic equation f(x)=x^2 - 6x + 14. We want to express it as f(x)=(x-h)^2 + k.
First, let's complete the square. We take half of the coefficient of x (-6) and square it: (-6/2)^2 = 9.
So, we can re-write the equation as: f(x) = (x^2 - 6x + 9) + 14 - 9.
Now, we have a perfect square trinomial, (x^2 - 6x + 9), which can be factored as (x - 3)^2.
Substituting that back into the equation, we get: f(x) = (x - 3)^2 + 14 - 9.
Simplifying further, we have: f(x) = (x - 3)^2 + 5.
Therefore, h = 3 and k = 5.
b. Since we've expressed the equation in the form f(x)=(x-h)^2 + k, we know that the vertex of the parabola is given by the coordinates (h, k).
In this case, the coordinates of the vertex are (3, 5).
a. To express f(x)=x^2 - 6x + 14 in the form f(x)=(x-h)^2 + k, we need to complete the square.
First, let's group the x terms together:
f(x) = x^2 - 6x + 14
Next, we will add and subtract the square of half the coefficient of the x term (which is (-6)/2 = -3):
f(x) = (x^2 - 6x + (-3)^2) - (-3)^2 + 14
Simplifying the expression inside the parentheses:
f(x) = (x^2 - 6x + 9) - 9 + 14
Combine like terms:
f(x) = (x - 3)^2 + 5
So, f(x) in the desired form is f(x) = (x - 3)^2 + 5.
b. From the expression f(x) = (x - 3)^2 + 5, we can see that the vertex of the parabola is located at point (h, k) = (3, 5).
Therefore, the coordinates of the vertex of the parabola y = x^2 - 6x + 14 are (3, 5).
a. To express the quadratic function f(x) = x^2 - 6x + 14 in the form f(x) = (x - h)^2 + k, where h and k are to be determined, we need to complete the square.
Step 1: Take the coefficient of x, which is -6, and divide it by 2: -6/2 = -3. This gives us the value for h, which is the x-coordinate of the vertex.
Step 2: Square the value of h: (-3)^2 = 9. This gives us the value for k, which is the y-coordinate of the vertex.
Step 3: Rewrite f(x) by adding and subtracting the value of k (9) inside the parentheses: f(x) = x^2 - 6x + 9 + 14 - 9.
Step 4: Rearrange the terms: f(x) = (x^2 - 6x + 9) + 14 - 9.
Step 5: Factor the perfect square trinomial within the parentheses: f(x) = (x - 3)^2 + 5.
Therefore, the quadratic function f(x) = x^2 - 6x + 14 can be expressed in the form f(x) = (x - 3)^2 + 5.
b. The coordinates of the vertex of the parabola equation y = x^2 - 6x + 14 can be determined using the completed square form obtained in part (a).
In the expression f(x) = (x - 3)^2 + 5, the vertex has coordinates (h, k). Therefore, the vertex of the parabola is located at the point (3, 5).
Hence, the coordinates of the vertex of the parabola equation y = x^2 - 6x + 14 are (3, 5).
f(x) = x^2 - 6x + 9 + 5
= (x-3)^2 + 5