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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