1. **Problem:** Given the base function $f(x) = x^2$, analyze the vertical translations for the functions: a. $g(x) = f(x) - 1 = x^2 - 1$ b. $y = x^2 + 1$ 2. **Formula and rules:** - The base quadratic function is $f(x) = x^2$. - Vertical translation means adding or subtracting a constant $k$ to $f(x)$: $y = f(x) + k$. - If $k > 0$, the graph shifts up by $k$ units. - If $k < 0$, the graph shifts down by $|k|$ units. - The domain of any quadratic function is all real numbers: $(-\infty, \infty)$. - The range depends on the vertex's $y$-value and the parabola's direction (upward here). 3. **Part a: $g(x) = x^2 - 1$** - The graph shifts down by 1 unit. - Vertex moves from $(0,0)$ to $(0,-1)$. - Domain remains $(-\infty, \infty)$. - Range is all $y$ such that $y \geq -1$. 4. **Part b: $y = x^2 + 1$** - The graph shifts up by 1 unit. - Vertex moves from $(0,0)$ to $(0,1)$. - Domain remains $(-\infty, \infty)$. - Range is all $y$ such that $y \geq 1$. **Final answers:** - a. Domain: $(-\infty, \infty)$ Range: $[-1, \infty)$ - b. Domain: $(-\infty, \infty)$ Range: $[1, \infty)$