Sketch the graph of each function. 1) \( y=-(x-2)^{2}-2 \)

We start with the function   y = –(x – 2)² – 2. Step 1. Rewrite in vertex form.  The function is already in the form   y = a(x – h)² + k, where a = –1, h = 2, and k = –2. This tells us:  • The vertex is at (h, k) = (2, –2).  • Because a = –1 (negative), the parabola opens downward. Step 2. Determine the axis of symmetry.  The axis of symmetry is the vertical line passing through the vertex, which is   x = 2. Step 3. Find the y-intercept.  Set x = 0:   y = –(0 – 2)² – 2 = –(4) – 2 = –6.  So, the y-intercept is at (0, –6). Step 4. Find additional points for accuracy.  Choose values symmetric about the vertex. For example:  For x = 1:   y = –(1 – 2)² – 2 = –(1) – 2 = –3.  So, (1, –3).  For x = 3 (symmetric to x = 1):   y = –(3 – 2)² – 2 = –(1) – 2 = –3.  So, (3, –3).  You can also evaluate at x = 4:   y = –(4 – 2)² – 2 = –(4) – 2 = –6.  So, (4, –6). Step 5. Sketching the graph.  • Plot the vertex at (2, –2).  • Draw the vertical line x = 2 as the axis of symmetry.  • Plot the y-intercept (0, –6) along with the symmetric point (4, –6) on the other side of the axis.  • Plot (1, –3) and (3, –3) which lie above the points just plotted.  • Draw a smooth downward-opening parabola through these points. Step 6. Note on x-intercepts.  Set y = 0 to find x-intercepts:   0 = –(x – 2)² – 2 ⟹ –(x – 2)² = 2 ⟹ (x – 2)² = –2.  Since (x – 2)² cannot be negative for real numbers, there are no real x-intercepts. Summary:  • Vertex: (2, –2)  • Axis of symmetry: x = 2  • y-intercept: (0, –6)  • Additional points: (1, –3) and (3, –3)  • The parabola opens downward and does not cross the x-axis. You can now sketch the graph using these key points and the overall shape.