15. \( y=-\frac{2}{3}|x-3|+4 \) Vertex: \( \frac{(3,4)}{\text { Domain: Al| real } \ddagger} \frac{(-00,4]}{\text { Range: }} \)

Let's analyze the given function step by step. 1. **Identify the function**: The function is given as \[ y = -\frac{2}{3}|x-3| + 4 \] 2. **Vertex**: The vertex of the absolute value function \( y = a|x-h| + k \) is at the point \( (h, k) \). In this case, \( h = 3 \) and \( k = 4 \), so the vertex is indeed \( (3, 4) \). 3. **Domain**: The domain of the function is all real numbers since there are no restrictions on \( x \). Therefore, the domain is: \[ \text{Domain: } (-\infty, \infty) \] 4. **Range**: To find the range, we need to analyze the behavior of the function. The absolute value function \( |x-3| \) is always non-negative, meaning \( |x-3| \geq 0 \). Therefore, the term \( -\frac{2}{3}|x-3| \) will always be less than or equal to 0. The maximum value of \( y \) occurs when \( |x-3| = 0 \) (i.e., at the vertex), which gives: \[ y = -\frac{2}{3}(0) + 4 = 4 \] As \( |x-3| \) increases, \( y \) decreases. Thus, the minimum value of \( y \) approaches negative infinity as \( |x-3| \) increases. Therefore, the range is: \[ \text{Range: } (-\infty, 4] \] In summary, we have: - Vertex: \( (3, 4) \) - Domain: \( (-\infty, \infty) \) - Range: \( (-\infty, 4] \)