12) Use your graphing calculator to find the key features [y-int, \( x \)-ints, and vertex] and then rewrite the function in both factored and vertex form. \[ h(x)=-2 x^{2}+14 x-16.5 \]

We start with the quadratic function   h(x) = –2x² + 14x – 16.5. Step 1. Find the y-intercept.   Set x = 0:    h(0) = –2(0)² + 14(0) – 16.5 = –16.5.   So, the y-intercept is (0, –16.5). Step 2. Find the vertex.   The x-coordinate for the vertex is given by –b/(2a). Here, a = –2 and b = 14.    x = –14 / (2 × –2) = –14 / –4 = 3.5  (or 7/2).   Now, find the y-coordinate by substituting x = 3.5 into h(x):    h(3.5) = –2(3.5)² + 14(3.5) – 16.5     = –2(12.25) + 49 – 16.5     = –24.5 + 49 – 16.5 = 8.   Thus the vertex is (3.5, 8) or (7/2, 8). Step 3. Find the x-intercepts.   The x-intercepts are found by setting h(x) equal to 0:    –2x² + 14x – 16.5 = 0.   Multiply through by –1:    2x² – 14x + 16.5 = 0,   and divide by 2 for simplicity:    x² – 7x + 8.25 = 0.   Using the quadratic formula, x = [7 ± √(7² – 4(1)(8.25)]/2, compute the discriminant:    Δ = 49 – 33 = 16.   Then,    x = [7 ± 4] / 2.   So, the two solutions are:    x = (7 + 4)/2 = 11/2 = 5.5  and  x = (7 – 4)/2 = 3/2 = 1.5.   Thus, the x-intercepts are (1.5, 0) and (5.5, 0). Step 4. Write the function in factored form.   Since the x-intercepts are x = 1.5 (or 3/2) and x = 5.5 (or 11/2), we can write:    h(x) = –2 (x – 1.5)(x – 5.5)   or, equivalently, in fractional form:    h(x) = –2 (x – 3/2)(x – 11/2). Step 5. Write the function in vertex form.   Using the vertex (3.5, 8), the vertex form is given by:    h(x) = a (x – h)² + k,   so here:    h(x) = –2 (x – 3.5)² + 8  or  h(x) = –2 (x – 7/2)² + 8. Summary of Key Features:   • y-intercept: (0, –16.5)   • x-intercepts: (1.5, 0) and (5.5, 0)   • vertex: (3.5, 8) Final Answers:   Factored form:  h(x) = –2 (x – 1.5)(x – 5.5)  or  h(x) = –2 (x – 3/2)(x – 11/2)   Vertex form:   h(x) = –2 (x – 3.5)² + 8  or  h(x) = –2 (x – 7/2)² + 8