Question

What is the equation of the blue graph?

Answer 1

Option C;
G(x) = (x - 1)² - 3
This form of a quadratic equation is attained by 'completing the square';
This form tells us the coordinates of the vertex of a quadratic, i.e. the sole turning/stationary point or the highest/lowest point of the quadratic graph;
In general, for any quadratic that you complete the square with, you will get:
(x + a)² + b, for which the vertex is (-a, b).
In the case of G(x), we can see from the graph that the vertex (lowest point) is: (1, -3)
Therefore the completed the square form of the equation we want is going to be: (x - 1)² - 3.

Answer 2

C. The vertex of the parabola is at (1, -3). Using this information:

a) plug in 1 for x. Since G(1)=-3 (because of the point given, (1,-3) ), we can determine that C is the only function that fits the criteria.

b) You can also think about it in terms of shifting. If you have an equation like this, or if you are finding the center of a circle or anything with a similar equation, (x-x1)^2 gives you (in this scenario) the x value of the vertex of the parabola (x1). It's where x=x1 so when you subtract x1 you get 0. Now, if you see the answers A and D given, and you see (x+1)^2, think of it like (x-(-1))^2. Since the vertex of this parabola is at x=1 and not x=-1, you can automatically rule out those two options.