1. **State the problem:** We analyze the function $f(x) = |x - 4| + 2$ to determine its key features such as intervals of increase/decrease, intercepts, domain, and range. 2. **Recall the absolute value function properties:** The graph of $y = |x - h| + k$ is a V-shaped graph with vertex at $(h, k)$, opening upwards. 3. **Identify the vertex:** Here, the vertex is at $(4, 2)$. 4. **Determine intervals of increase and decrease:** - For $x < 4$, $f(x) = -(x - 4) + 2 = -x + 6$, which is a decreasing linear function. - For $x > 4$, $f(x) = (x - 4) + 2 = x - 2$, which is an increasing linear function. 5. **Check the x-intercept:** Solve $f(x) = 0$: $$|x - 4| + 2 = 0 \implies |x - 4| = -2$$ Since absolute value cannot be negative, there are no x-intercepts. 6. **Check the y-intercept:** Evaluate $f(0)$: $$f(0) = |0 - 4| + 2 = 4 + 2 = 6$$ So the y-intercept is $(0, 6)$. 7. **Determine the domain:** The absolute value function is defined for all real numbers, so domain is $(-\infty, \infty)$. 8. **Determine the range:** The minimum value is at the vertex $y=2$, and since the graph opens upward, range is $[2, \infty)$. **Final answers:** - The function is **increasing** on the interval $\{x | x > 4\}$. - The function is **not decreasing** on $\{x | x > 4\}$. - The function has **no x-intercepts**. - The y-intercept is $(0, 6)$. - The domain is $(-\infty, \infty)$. - The range is $y \geq 2$.