- The function \(f(x) = 2(x + 3)^2 - 1\) is in vertex form.
- The parabola opens upward since \(a = 2 > 0\).
- The vertex of the parabola is at \((-3, -1)\).
- Therefore, the correct options are: \(\boxed{D, F, G}\).
Explanation
- Identifying the Form of the Function
The function \(f(x) = 2(x + 3)^2 - 1\) is a quadratic function. To identify its form and key characteristics, we compare it to the standard definitions:
- Standard form: \(f(x) = ax^2 + bx + c\).
- Intercept form: \(f(x) = a(x - p)(x - q)\).
- Vertex form: \(f(x) = a(x - h)^2 + k\).
Comparing our function \(f(x) = 2(x + 3)^2 - 1\) to these forms, we can see that it matches the vertex form, where \(a = 2\), \(h = -3\), and \(k = -1\). Thus, the function is in vertex form (Option F).
- Determining the Direction of the Parabola
The parameter \(a\) in the vertex form \(f(x) = a(x - h)^2 + k\) determines the orientation of the parabola. Since \(a = 2\), and \(2 > 0\), the parabola opens upward (Option D). If \(a\) were negative, it would open downward.
- Finding the Vertex
The vertex of a parabola in vertex form \(f(x) = a(x - h)^2 + k\) is given by the point \((h, k)\). For our function \(f(x) = 2(x - (-3))^2 + (-1)\), we identify \(h = -3\) and \(k = -1\). Therefore, the vertex is at \((-3, -1)\) (Option G).
- Analyzing Intercepts and Final Conclusion
To determine the y-intercept, we evaluate the function at \(x = 0\):
\[f(0) = 2(0 + 3)^2 - 1 = 2(3^2) - 1 = 2(9) - 1 = 18 - 1 = 17.\]
So the y-intercept is \((0, 17)\). Option H suggests the y-intercept is \((0, -1)\), which is incorrect.
To determine the x-intercepts, we solve for \(f(x) = 0\):
\[2(x + 3)^2 - 1 = 0 \implies (x + 3)^2 = \frac{1}{2} \implies x + 3 = \pm\frac{1}{\sqrt{2}} \implies x = -3 \pm\frac{\sqrt{2}}{2}.\]
These are not \((-1, 0)\), so Option E is incorrect.
Thus, the correct descriptions are vertex form (F), parabola opens up (D), and vertex: (-3, -1) (G).
Examples
Quadratic functions are widely used in modeling trajectories, such as the path of a ball kicked or a projectile launched. For example, if the height of a drone at time \(t\) is modeled by a function like \(f(t) = 2(t - 3)^2 + 10\), knowing the vertex \((3, 10)\) allows the operator to quickly identify the minimum height of the drone and when it occurs. Similarly, architects use parabolic curves in designing arches for bridges or roofs to distribute weight efficiently and create aesthetically pleasing structures, where understanding the vertex and orientation is crucial for construction.