The given quadratic function is:

$f(x) = 2(x + 2)^2 - 1$

Step 1: Determine the Direction of the Parabola

The general form of a quadratic function is $y = a(x-h)^2 + k$, where:

• $a$ controls the direction and the "stretch" or "compression" of the graph.
• If $a > 0$, the parabola opens upward.
• If $a < 0$, the parabola opens downward.

In this case, $a = 2$, which is positive. Therefore, the parabola opens upward.

Step 2: Determine the Stretch or Compression

• If $|a| > 1$, the parabola is stretched taller compared to the parent function $y = x^2$.
• If $0 < |a| < 1$, the parabola is compressed shorter.

Since $|a| = 2 > 1$, the parabola is stretched taller by a factor of 2.

Conclusion:

The parabola opens upward and is stretched taller by a factor of 2.

The correct answer is:

• "The parabola opens upward and is stretched taller by a factor of 2."

Would you like more details or have any questions? Here are some related questions:

1. How do you find the vertex of the quadratic function?
2. What effect does the value $h$ have in the vertex form of a parabola?
3. How does shifting a parabola horizontally and vertically work?
4. Can a quadratic function have a negative $a$ but still be stretched?
5. What is the parent quadratic function and how does it look?

Tip: In the vertex form $f(x) = a(x-h)^2 + k$, the vertex is $(h, k)$.