1a)Horizontal shift left by 2 units (x + 2).
Vertical shift down by 4 units (- 4).
b) The vertex is (-2, -4)
c) g(x), the parabola opens upward with vertex (-2,-4).
2) Downward facing parabola.
Upward facing parabola, shifted up by 1 unit.
Upward facing parabola, shifted left by 3 units.
3) If ( a > 0 ), the parabola opens upward;
if ( a < 0 ), it opens downward.
What is a parabola?
Given
g(x) = (x + 2)² - 4, parent function p(x) = x²
a) Transformations from p(x) = x²:
Horizontal shift left by 2 units (x + 2).
Vertical shift down by 4 units (- 4).
b) Vertex of the Parabola:
y = a(x - h)² + k
Where
(h, k) is the vertex of the parabola.
a is a constant that determines the direction and width of the parabola.
If ( a > 0 ), the parabola opens upward;
if ( a < 0 ), it opens downward.
Compare with the general equation
h = -2, k = -4
The vertex is (-2, -4)
c)Graph of y = g(x), the parabola opens upward because a = 1 which is greater than 0.
2.Parabolas and Equations:
f(x) = -x²: Downward facing parabola.
g(x) = x² + 1: Upward facing parabola, shifted up by 1 unit.
k(x) = (x + 3)²: Upward facing parabola, shifted left by 3 units.
3.Parabola Direction (Opening Up or Down):
If the coefficient of the x² term is positive, the parabola opens upward.
If the coefficient of the x² term is negative, the parabola opens downward.