Describe how the given function can be obtained from one of the basic graphs. Then graph the function. $$ g(x)=(x+1)^{2}-2 $$

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1. **Identify the Basic Graph**

The given function is  
   $$
   g(x) = (x+1)^2 - 2.
   $$  
   Recognize that the basic function here is  
   $$
   f(x) = x^2,
   $$  
   which is the standard parabola.

2. **Determine the Transformations**

- **Horizontal Translation:**  
     The term $$(x+1)^2$$ indicates a shift of the basic graph $$f(x) = x^2$$ to the left by 1 unit.  
     
   - **Vertical Translation:**  
     The subtraction of 2, $$-2$$, indicates a downward shift of the entire graph by 2 units.

3. **Locate the Vertex**

For the function $$g(x) = (x+1)^2 - 2$$, the vertex of the parabola is obtained by taking the translation into account.  
   - The vertex of $$f(x)=x^2$$ is $$(0,0)$$.  
   - After shifting left by 1 and down by 2, the vertex becomes  
     $$
     (-1, -2).
     $$

4. **Graph the Function**

- **Shape:**  
     The graph is a parabola that opens upward because the coefficient of the squared term is positive.
     
   - **Vertex:**  
     The vertex is at $$(-1, -2)$$, which is the lowest point on the graph.
     
   - **Axis of Symmetry:**  
     The vertical line $$x = -1$$ is the axis of symmetry.
     
   - **Key Points:**
     - When $$x = 0$$:
       $$
       g(0) = (0+1)^2 - 2 = 1 - 2 = -1,
       $$
       so the point $$(0, -1)$$ lies on the graph.
     - When $$x = -2$$:
       $$
       g(-2) = (-2+1)^2 - 2 = (-1)^2 - 2 = 1 - 2 = -1,
       $$
       so the point $$(-2, -1)$$ is also on the graph.
     - These points confirm the symmetry about the line $$x = -1$$.

5. **Conclusion: Transformation Process**

The function $$g(x) = (x+1)^2 - 2$$ is obtained from the basic quadratic function $$f(x) = x^2$$ by:
   - Shifting the graph to the left by 1 unit
   - Shifting the resulting graph downward by 2 units

6. **Sketching the Graph**

To sketch the graph:
   - Plot the vertex at $$(-1, -2)$$.
   - Draw the axis of symmetry along the line $$x = -1$$.
   - Plot the points $$(0, -1)$$ and $$(-2, -1)$$ to show symmetry.
   - Sketch the parabola opening upward, passing through these points.

The final graph is a parabola with vertex at $$(-1, -2)$$, opening upward, and symmetric about the line $$x = -1$$.
To graph the function $$ g(x) = (x+1)^2 - 2 $$, start with the basic parabola $$ f(x) = x^2 $$. Shift it left by 1 unit and down by 2 units. The vertex is at $$(-1, -2)$$, and the parabola opens upward. Plot the vertex and key points like $$(0, -1)$$ and $$(-2, -1)$$, then sketch the symmetric parabola.

Describe how the given function can be obtained from one of the basic graphs. Then graph the function. \( g(x)=(x+1)^{2}-2 \)

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