Parabola Graph F83338

1. **State the problem:** We are given the function $$f(x) = 2(x - 2)^2 - 6$$ and asked to sketch its graph. 2. **Identify the type of function:** This is a quadratic function in vertex form $$f(x) = a(x - h)^2 + k$$ where $$a = 2$$, $$h = 2$$, and $$k = -6$$. 3. **Important properties:** - The parabola opens upward because $$a = 2 > 0$$. - The vertex is at $$(h, k) = (2, -6)$$. - The axis of symmetry is the vertical line $$x = 2$$. 4. **Sketching the graph:** - Start by plotting the vertex at $$(2, -6)$$. - Since $$a = 2$$, the parabola is narrower than the standard parabola $$y = x^2$$. - To find additional points, choose values of $$x$$ near 2: - For $$x = 1$$: $$f(1) = 2(1 - 2)^2 - 6 = 2( -1)^2 - 6 = 2(1) - 6 = -4$$ - For $$x = 3$$: $$f(3) = 2(3 - 2)^2 - 6 = 2(1)^2 - 6 = 2 - 6 = -4$$ - Points $$(1, -4)$$ and $$(3, -4)$$ are symmetric about the axis of symmetry. 5. **Summary:** The graph is a parabola opening upward with vertex at $$(2, -6)$$, axis of symmetry $$x = 2$$, passing through points $$(1, -4)$$ and $$(3, -4)$$. **Final answer:** The parabola is centered at $$(2, -6)$$, opens upward, and is narrower than the standard parabola due to $$a=2$$.