See below
Step-by-step explanation:
The vertex form of a parabola is y = a(x - h)^2 + k
If the leading coefficient is positive, the parabola opens upward, and (h, k) represents the vertex, or turning point.
1a. The leading coefficient is 4, which is positive, meaning the parabola opens upwards. From the vertex form, we can say the vertex(turning point) is (2, 7).
1b. The leading coefficient is -3, which is negative, meaning the parabola opens downwards. From the vertex form, we can say the vertex(turning point) is (-6, 4).
1c. The leading coefficient is -1, which is negative, meaning the parabola opens downwards. From the vertex form, we can say the vertex(turning point) is (-4, -3).
To write the vertex form of a parabola, we need to complete the square. A perfect square trinomial is in the form a^2 + 2ab + b^2. In these cases we are already given a^2 and 2ab.
2a. We can say x^2 corresponds to a^2 because it is a square and 2ab corresponds to 6x. From this, we can say a = x. Now, we can plug that in:
2ab = 6x
2(x)b = 6x
b = 3
Therefore, the last term b^2, is 3^2 or 9. We can add 9 and subtract nine to keep the equation equivalent:
y = x^2 + 6x + 9 - 9 - 2
Now, we can factor the square into (a + b)^2 and simplify:
y = (x + 3)^2 - 11
From this form, the vertex(turning point) will be (-3, -11). Since the leading coefficient is 1, which is positive, the parabola opens upwards, meaning the turning point is a minimum.
2b. We can say x^2 corresponds to a^2 because it is a square and 2ab corresponds to -2x. From this, we can say a = x. Now, we can plug that in:
2ab = -2x
2(x)b = -2x
b = -1
Therefore, the last term b^2, is (-1)^2 or 1. We can add 1 and subtract nine to keep the equation equivalent:
y = x^2 - 2x + 1 - 1 + 11
Now, we can factor the square into (a + b)^2 and simplify:
y = (x - 1)^2 + 10
From this form, the vertex(turning point) will be (1, 10). Since the leading coefficient is 1, which is positive, the parabola opens upwards, meaning the turning point is a minimum.
