$$ \left. \begin{array} { l } { y = | x - 6 | \quad y = x ^ { 2 } + 5 \quad y = - x ^ { 2 } + 5 \quad y = | x | + 6 \quad y = - ( x + 5 ) ^ { 2 } } \ { y = - | x + 6 | \quad y = - | x | + 6 \quad y = - 6 | x | \quad y = ( x - 5 ) ^ { 2 } \quad y = - 5 x ^ { 2 } } \ { y = 5 x ^ { 2 } \quad y = 6 | x | } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { y = | x - 6 | \quad y = x ^ { 2 } + 5 \quad y = - x ^ { 2 } + 5 \quad y = | x | + 6 \quad y = - ( x + 5 ) ^ { 2 } } \ { y = - | x + 6 | \quad y = - | x | + 6 \quad y = - 6 | x | \quad y = ( x - 5 ) ^ { 2 } \quad y = - 5 x ^ { 2 } } \ { y = 5 x ^ { 2 } \quad y = 6 | x | } \end{array} ight. $$

UpStudy AI · Accepted Answer

1. $$ y = |x-6| $$  
 – This is an absolute‐value function.  
 – **Vertex:** The expression inside the absolute value is zero at $$ x = 6 $$; hence the vertex is at $$(6,0)$$.  
 – **Domain:** All real numbers, $$\mathbb{R}$$.  
 – **Range:** $$ y \ge 0 $$.  
 – **Shape:** “V” shape opening upward.

2. $$ y = x^2 + 5 $$  
 – This is a parabola opening upward.  
 – **Vertex:** In the form $$ y = (x-0)^2+5 $$, the vertex is $$(0,5)$$.  
 – **Domain:** $$\mathbb{R}$$.  
 – **Range:** $$ y \ge 5 $$.

3. $$ y = -x^2 + 5 $$  
 – This is a downward-opening parabola.  
 – **Vertex:** In the form $$ y = -(x-0)^2+5 $$, the vertex is at $$(0,5)$$.  
 – **Domain:** $$\mathbb{R}$$.  
 – **Range:** $$ y \le 5 $$.

4. $$ y = |x| + 6 $$  
 – This is an absolute‐value function shifted upward.  
 – **Vertex:** The minimum occurs at $$ x = 0 $$ giving the vertex $$(0,6)$$.  
 – **Domain:** $$\mathbb{R}$$.  
 – **Range:** $$ y \ge 6 $$.  
 – **Shape:** “V” shape opening upward.

5. $$ y = -(x+5)^2 $$  
 – This is a downward-opening parabola.  
 – **Vertex:** Writing it as $$ y = -[(x+5)^2] $$, the vertex is at $$ x = -5 $$ with $$ y = 0 $$; so $$(-5,0)$$.  
 – **Domain:** $$\mathbb{R}$$.  
 – **Range:** $$ y \le 0 $$.

6. $$ y = -|x+6| $$  
 – This is a “V” shaped absolute‐value function reflected vertically.  
 – **Vertex:** The expression inside is zero when $$ x = -6 $$; vertex at $$(-6,0)$$.  
 – **Domain:** $$\mathbb{R}$$.  
 – **Range:** $$ y \le 0 $$.

7. $$ y = -|x| + 6 $$  
 – This function takes the absolute value, reflects it vertically and shifts it upward.  
 – **Vertex:** Occurs at $$ x = 0 $$ with $$ y = 6 $$; vertex $$(0,6)$$.  
 – **Domain:** $$\mathbb{R}$$.  
 – **Range:** $$ y \le 6 $$.  
 – **Shape:** Inverted “V” shape (an “∧” shape) with maximum at $$(0,6)$$.

8. $$ y = -6|x| $$  
 – This is a vertically scaled, reflected absolute‐value function.  
 – **Vertex:** At $$ x = 0 $$, $$ y = 0 $$; so vertex $$(0,0)$$.  
 – **Domain:** $$\mathbb{R}$$.  
 – **Range:** $$ y \le 0 $$.

9. $$ y = (x-5)^2 $$  
 – This is an upward-opening parabola.  
 – **Vertex:** In the form $$ y = (x-5)^2 $$, the vertex is at $$(5,0)$$.  
 – **Domain:** $$\mathbb{R}$$.  
 – **Range:** $$ y \ge 0 $$.

10. $$ y = -5x^2 $$  
 – This is a downward-opening parabola with vertical stretch.  
 – **Vertex:** In the form $$ y = -5(x-0)^2 $$, the vertex is at $$(0,0)$$.  
 – **Domain:** $$\mathbb{R}$$.  
 – **Range:** $$ y \le 0 $$.

11. $$ y = 5x^2 $$  
 – This is an upward-opening parabola with vertical stretch.  
 – **Vertex:** Written as $$ y = 5(x-0)^2 $$, the vertex is at $$(0,0)$$.  
 – **Domain:** $$\mathbb{R}$$.  
 – **Range:** $$ y \ge 0 $$.

12. $$ y = 6|x| $$  
 – This is an absolute‐value function with vertical stretching.  
 – **Vertex:** At $$ x = 0 $$, $$ y = 0 $$; so vertex $$(0,0)$$.  
 – **Domain:** $$\mathbb{R}$$.  
 – **Range:** $$ y \ge 0 $$.

Each function has been analyzed with respect to its vertex, domain, range, and general shape.
Each function is described by its type (absolute value, parabola), vertex, domain, range, and general shape.

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