Graph (V. Hard) (10x2pts)

The answer of this question is G

In this question,

x=0, the answer is 1
x=0.5, the answer is 1 also.

So, we sudst. x=0 an x=0.5 into the equation.

In A, x=0.5, answer is not 1

In B, x=0.5, as same as A, not 1

In C, x=0.5, answer is not 1

In D, x=0, answer is not 1

In E, x=0, answer is not 1

In F, x=0, answer is not 1

But G, x=0, answer is 1
x=0.5,answer is also 1

So, according to {(Can choose one only)},
G is the answer.

I have to tell you that
　NONE of the options are appropriate.
You can check it by substituting several "observable" points on the parabola e.g. (-1, 1.5) and (-1.5, 2.5) to get the y-coordinate.

The correct answer is x^2+0.5x+1.
Here displays the full solution to the question:


From the graph, you should identify the following:

1) The graph is a parabola (major axis // y-axis).
⇒ It represents a quadratic function.
∴ G, H eliminated
*2) The graph does not intersect the x-axis.
⇒ △<0
∴ A, B, F eliminated
*3) y-intercept of the graph = 1
⇒ The constant term is 1. (no elimination)

*Step 2, 3 may be skipped for FASTER calculation of the answer.

Note that when the graph is translated along the -ve y-axis by 1 unit, there will be two distinct real roots of 0 and -0.5.

　　Ax^2+Bx+C = A(x-root1)(x-root2)

　y-1 = A(x-0)(x+0.5)　where A is a real number
　y = A(x^2+0.5x)+1　...(1)

To find out A, substitute a point on the parabola (-1, 1.5) into (1): (x^2+0.5x≠0)
　1.5 = A[(-1)^2+0.5(-1)]+1
　1.5 = 0.5A+1
　A = 1
∴The equation is x^2+0.5x+1.


The nearest but INCORRECT answer is E, 4x^2+2x+1.
If we compare it with equation (1), A=4, so the roots at (-0.5, 1) and (0, 1) will return a positive test result.
However, when we test the point (-1, 1.5) with y=4x^2+2x+1,
we will get
　LHS = 1.5
　RHS = 4(-1)^2+2(-1)+1
　　 　= 4-2+1
　　　 = 3 ≠ LHS

The graphs are plotted as below:
http://xs132.xs.to/xs132/08434/vhgraph001519.gif

When x = 0, y = 1. So the possibilities are C, D and E only.
The axis of symmetry is x = -1/4. Since axis of symmetry is x = -b/2a, that is
-b/2a = -1/4
b/a = 1/2
So the answer is E.