Zimmerman Allan
09/02/2024 · Elementary School
Sketch on the same set of axes the graphs of \( f(x)=-2 x^{2}-4 x+6 \) and \( g(x)=-2 \cdot 2^{x-1}+1 \) Clearly indicate all intercepts with the axes, turning point(s) and asymptote(s).
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To sketch the graphs of \( f(x) = -2x^{2} - 4x + 6 \) and \( g(x) = -2 \cdot 2^{x-1} + 1 \) on the same set of axes:
1. **For \( f(x) \):**
- **Vertex:** \((-1, 8)\)
- **Y-intercept:** \((0, 6)\)
- **X-intercepts:** \((1, 0)\) and \((-3, 0)\)
- **Axis of Symmetry:** \( x = -1 \)
2. **For \( g(x) \):**
- **Horizontal Asymptote:** \( y = 1 \)
- **Y-intercept:** \((0, 0)\)
- **No Turning Points**
3. **Combined Graph:**
- Both graphs intersect at \((0, 0)\).
- \( f(x) \) is a downward-opening parabola with its vertex at \((-1, 8)\).
- \( g(x) \) is a decreasing exponential curve approaching \( y = 1 \) as \( x \) decreases and dropping to \( -\infty \) as \( x \) increases.
Ensure to plot all intercepts and asymptotes clearly on the same set of axes.
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