Which of the following sentences is true about the graphs of?

Question

y- 3(x-5)^2 +1 and y= 3(x+5)^2 + 1?

a. Their vertices are maximums.
b. The graphs have the same shape with different vertices.
c. The graphs have different shapes with different vertices.
d. One graph has a vertex that is a maximum, while the other graph has a vertex that is a minimum.

What is meant by a vertex that is a "maximum", or a "minimum"? I've simplified both equations down to:
y=3x^2- 30x+ 76 and
y= 3x^2+ 30x+ 76; can I predict how the graph will turn out with just the b terms?

Answer 1

The 2 equations you have are already in the best form to see what they look like.

The "vertex form" of a parabola (opening up or down) is:

y = a(x - h)² + k

The vertex of the parabola is (h, k).

If a is positive, the parabola opens upward, and the vertex is a minimum.

If a is negative, the parabola opens downward, and the vertex is a maximum.

Also, the "bigger" a is, the faster the parabola grows, and the "skinnier" it will look. The "smaller" a is, the slower the parabola grows.

You have two parabolas that are almost identical, but have different vertices:

y = 3(x - 5)² + 1              vertex (5, 1), a = 3

y = 3(x + 5)² + 1             vertex (-5, 1), a = 3

a = 3 for both, so both parabolas open upwards and their vertices are minimums. This makes letter a) false.

Also since they have the same a, they both increase at the same rate, i.e. they have the same shape. This makes letter c) false.

And d is clearly false since a = 3, for both; the only way d) could be true would be if a was positive for one and negative for the other.

So the only correct answer is b.

Answer 2

the answer should be B, since when you have both of the equations in vertex form [y = a(x - h)^2 +k]. if only h and k changes that means only the vertex is different and everything else is the same.