Function Transformations Aab1F0

1. The problem asks to sketch the graphs of $Q(x)=2.W(x)$, $R(x)=-W(x)$, and $S(x)=-2.W(x)$ given the graph of $F(x) = 2x^3 - 3x^2 + 1$. 2. First, we need to understand what $W(x)$ represents. Since the problem does not define $W(x)$ explicitly, we assume $W(x)$ is the function $F(x)$ or related to it. 3. Assuming $W(x) = F(x) = 2x^3 - 3x^2 + 1$, we analyze each function: - $Q(x) = 2.W(x) = 2 imes (2x^3 - 3x^2 + 1) = 4x^3 - 6x^2 + 2$ - $R(x) = -W(x) = -(2x^3 - 3x^2 + 1) = -2x^3 + 3x^2 - 1$ - $S(x) = -2.W(x) = -2 imes (2x^3 - 3x^2 + 1) = -4x^3 + 6x^2 - 2$ 4. These transformations involve vertical scaling and reflection: - Multiplying by 2 in $Q(x)$ stretches the graph vertically by a factor of 2. - Multiplying by -1 in $R(x)$ reflects the graph of $W(x)$ about the x-axis. - Multiplying by -2 in $S(x)$ reflects and stretches the graph vertically by a factor of 2. 5. To sketch these graphs, start with the graph of $W(x)$ (or $F(x)$), then apply the transformations: - For $Q(x)$, multiply all y-values by 2. - For $R(x)$, multiply all y-values by -1. - For $S(x)$, multiply all y-values by -2. 6. The final expressions for the functions are: $$Q(x) = 4x^3 - 6x^2 + 2$$ $$R(x) = -2x^3 + 3x^2 - 1$$ $$S(x) = -4x^3 + 6x^2 - 2$$ These can be plotted or sketched accordingly.