Answer
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.
Solution
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 \(x = -1\).
Answered by UpStudy AI and reviewed by a Professional Tutor
Explain
Simplify this solution
