The image you've uploaded contains three mathematical functions that need to be analyzed in terms of their properties, including the parent function, transformations, domain, range, intercepts, extrema, and behavior. Let's go through each of the three problems one by one.

Problem 4:

The function is: $f(x) = \frac{1}{2}(x - 5)^2 - 3$

• Parent Function: The parent function is $f(x) = x^2$, which is a basic quadratic function.

• Transformations:

  1. Horizontal shift: The term $(x - 5)$ indicates a shift 5 units to the right.
  2. Vertical stretch: The coefficient $\frac{1}{2}$ causes a vertical compression by a factor of 2.
  3. Vertical shift: The term $-3$ indicates a shift 3 units down.

• Domain: The domain is all real numbers, $(-\infty, \infty)$, because quadratic functions are defined everywhere.

• Range: Since the parabola opens upwards and the vertex is at $(5, -3)$, the range is $[-3, \infty)$.

• Intercepts:

  1. y-intercept: To find this, set $x = 0$ in the equation. $f(0) = \frac{1}{2}(0 - 5)^2 - 3 = \frac{1}{2}(25) - 3 = 12.5 - 3 = 9.5$ So, the y-intercept is $(0, 9.5)$.
  2. x-intercepts: Set $f(x) = 0$ and solve for $x$. $\frac{1}{2}(x - 5)^2 - 3 = 0 \quad \Rightarrow \quad \frac{1}{2}(x - 5)^2 = 3 \quad \Rightarrow \quad (x - 5)^2 = 6$ $x - 5 = \pm \sqrt{6} \quad \Rightarrow \quad x = 5 \pm \sqrt{6}$ The x-intercepts are $5 + \sqrt{6}$ and $5 - \sqrt{6}$.

• Extrema: The vertex is the minimum point, which occurs at $(5, -3)$.

• End Behavior: As $x \to \infty$, $f(x) \to \infty$; and as $x \to -\infty$, $f(x) \to \infty$, since the parabola opens upwards.

Problem 5:

The function is: $f(x) = 3(x - 2)^2 + 1$

• Parent Function: The parent function is again $f(x) = x^2$.

• Transformations:

  1. Horizontal shift: $x - 2$ indicates a shift 2 units to the right.
  2. Vertical stretch: The coefficient $3$ stretches the graph vertically by a factor of 3.
  3. Vertical shift: The term $+1$ indicates a shift 1 unit upwards.

• Domain: The domain is all real numbers, $(-\infty, \infty)$.

• Range: Since the vertex is at $(2, 1)$ and the parabola opens upwards, the range is $[1, \infty)$.

• Intercepts:

  1. y-intercept: Set ( x =