Question

types of parabolas
$y = x^2 + 2x - 5$ | $y = -x^2 + 2x - 5$
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sketch:
coordinate axis
when \a\ is ______, the parabola opens __.
when it opens up, the graph has a ________.
this is also the vertex. | sketch:
coordinate axis
when \a\ is ______, the parabola opens __.
when it opens down, the graph has a ________.
this is also the vertex.

Explanation:

Step1: Analyze first parabola ($y=x^2+2x-5$)

For the standard quadratic form $y=ax^2+bx+c$, here $a=1$ (positive). A positive $a$ means the parabola opens upward, and upward-opening parabolas have a minimum point at the vertex.

Step2: Analyze second parabola ($y=-x^2+2x-5$)

For this quadratic, $a=-1$ (negative). A negative $a$ means the parabola opens downward, and downward-opening parabolas have a maximum point at the vertex.

Answer:

For $y = x^2 + 2x - 5$:

When "a" is positive, the parabola opens up.
When it opens UP, the graph has a minimum point
This is also the vertex.

For $y = -x^2 + 2x - 5$:

When "a" is negative, the parabola opens down.
When it opens DOWN, the graph has a maximum point
This is also the vertex.