Consider the quadratic relation y=2x2-4x+5
Convert to vertex form by completing the square.
Explain whether this relation has a maximum or a minimum.
To convert the quadratic relation to vertex form by completing the square, we first write the equation as:
y = 2(x^2 - 2x) + 5.
Next, we need to find the value that completes the square inside the parentheses. To do this, we take half of the coefficient of x (which is -2) and square it. Half of -2 is -1, and (-1)^2 = 1. Therefore, we add 1 inside the parentheses:
y = 2(x^2 - 2x + 1) + 5 - 2(1).
Simplifying further:
y = 2(x^2 - 2x + 1) + 5 - 2
y = 2(x - 1)^2 + 3
Now the quadratic relation is in vertex form, which is y = a(x - h)^2 + k.
The vertex form of the given relation is y = 2(x - 1)^2 + 3, where (h, k) represents the vertex of the parabola.
Regarding whether this relation has a maximum or minimum, we can determine it based on the coefficient of the x^2 term (a).
Since the coefficient of x^2 is positive (2), the parabola opens upwards, implying that it has a minimum value.
Therefore, the relation y = 2x^2 -4x + 5 has a minimum value.