Last question. use the table of values that represent points on the graph of a quadratic function. by determining the vertex and axis of symmetry, find the general form of the equation of the quadratic function. x = -2, -1, 0, 1, 2 y = 1, 0, 1, 4, 9

Examining the y-values shows that all of the given y-values are perfect squares. Further, the graph dips at first, reaches a minimum (-1,0) and then increases as does the second power of x.

This is sufficient info on which to base the conclusion that the vertex of this parabolic function is at (-1,0), and that the axis of symmetry (which goes thru the vertex) is x = -1.

The most basic form of the equation of a parabola is y = x^2. The graph is that of a parabola and opens up. The vertex is (0,0) and the axis of symmetry is x=0.

In this particular case, the vertex has been translated 1 unit to the left. The corresponding equation of this "new" parabola is y = (x+1)^2. Note that (2,9) (one of the points given in the table) satisfies this equation:

9 = (2+1)^2.

The desired equation is y = (x+1)^2.

3. parallelogram

step-by-step explanation:

we turn each of our sides into direction vectors

ab=b-a=(3 - 7, 1 - 0) = (-4, 1)

bc=c-b=(1,6)

cd=d-c=(4,-1)

da=a-d=(-1,-6)

we see how the opposite sides have negated direction vectors; this is characteristic of a parallelogram.   we have congruent sides ab=cd and bc=da with unequal lengths because 1^2+6^2 is different from (-4)^2+1^2.   so it's not a square or a rhombus.  

we check the dot product of direction vectors of adjacent sides.   we have

ab dot bc = -4(1) + 1(6) = 2

that's not zero so the sides are not perpendicular.   we've ruled out rectangle.

the direction vectors indicate two pairs of congruent parallel sides, so we choose parallelogram over trapezoid.