1. **Problem statement:** We are given three functions: - $f_b(x) = (x-b)^2$ - $f_k(x) = -2x^4 + 2k^2x^2$ - $f_a(x) = -8x + 16a$ We will analyze each function separately. --- 2. **Function $f_b(x) = (x-b)^2$** - This is a quadratic function representing a parabola. - The vertex form of a parabola is $f(x) = (x-h)^2 + c$, where $(h,c)$ is the vertex. - Here, the vertex is at $(b,0)$. - The parabola opens upwards because the coefficient of the squared term is positive. --- 3. **Function $f_k(x) = -2x^4 + 2k^2x^2$** - This is a quartic polynomial. - We can factor it as: $$f_k(x) = -2x^4 + 2k^2x^2 = 2x^2(k^2 - x^2)(-1) = -2x^2(x^2 - k^2)$$ - The roots are at $x=0$ and $x=\pm k$. - The leading term $-2x^4$ indicates the graph opens downward for large $|x|$. --- 4. **Function $f_a(x) = -8x + 16a$** - This is a linear function. - The slope is $-8$, so the line decreases as $x$ increases. - The y-intercept is $16a$. --- **Summary:** - $f_b(x)$ is a parabola with vertex at $(b,0)$ opening upwards. - $f_k(x)$ is a quartic with roots at $0$ and $\pm k$, opening downward at extremes. - $f_a(x)$ is a decreasing linear function with y-intercept $16a$.