Final answer:
To graph the parabola y = 2(x + 3)² - 4, the vertex form reveals the shifts: the parabola is translated 3 units to the left and 4 units down with a vertex at (3, -4). The value of 'a' being 2 creates an upward opening, narrower parabola.
Step-by-step explanation:
The equation y = 2(x + 3)² - 4 represents a parabola that is shifted from its standard position. To graph this parabola, recognize that it is in the vertex form y = a(x - h)² + k, where (h, k) is the vertex of the parabola, and a determines the width and direction of the opening.
To find the values of a, h, and k:
- a is the coefficient of the squared term, which is 2 in this case. It tells us the parabola opens upwards and is narrower than the standard y=x² because a > 1.
- h is the opposite sign of the value inside the parentheses with the x, which is -3. Thus, h = 3. This shifts the parabola 3 units to the left.
- k is the constant term added or subtracted at the end of the equation, which is -4. This shifts the parabola 4 units downwards.
The vertex of the parabola is at the point (h, k) = (3, -4). To graph this parabola, you would start by plotting the vertex and then sketching the curve, making sure it opens upwards and appearing narrower because of the a value.