This question requires the use of geometric concepts, specifically the Pythagorean theorem. To solve it, students must apply the theorem to find the length of the other leg.
Rewrite in standard form: (3x^2 + 12x - k = 0). Here, (a = 3), (b = 12), (c = -k). hard sat questions math
On the digital SAT, you have a built-in graphing calculator (Desmos). However, the hardest questions are designed to waste your time if you rely solely on graphing. This question requires the use of geometric concepts,
Axis of symmetry: ( x = 3 ) → vertex is (3, k). Points symmetric: (0,5) and (6,5) confirm symmetry. Write ( y = a(x-3)^2 + k ). Plug (0,5): ( 5 = 9a + k ). Plug (6,5): ( 5 = 9a + k ) (same eq). Need another point? Not given. But wait — they want ( a ) only. If vertex max, ( a<0 ). Hmm — maybe not enough info? Actually, this is a trick: points (0,5) and (6,5) same y → vertex x=3 means ( y = a(x-3)^2 + 5 ) (since at x=3, y=5? No, we don't know vertex y). Let's solve: From symmetry, vertex y = ? Plug x=3: ( y_v = 5 )? Not necessarily. Better: Use two points in standard form: (0,5): ( c=5 ). (6,5): ( 36a+6b+5=5 ) → ( 36a+6b=0 ) → ( 6a+b=0 ). Axis ( -b/(2a)=3 ) → ( -b=6a ) → ( b=-6a ). Substitute: ( 6a + (-6a) = 0 ) ok. So infinite a? No — they need a specific. Conclusion: This is a bad example unless vertex y given. So the real hard ones do give vertex or another point. Here, (a = 3), (b = 12), (c = -k)