ACT Math: Parabola Vertex Form vs. Standard Form—Convert and Interpret

Standard form: y=ax²+bx+c (useful for finding y-intercept and solving by factoring). Vertex form: y=a(x-h)²+k (vertex is (h,k), opens upward if a>0, downward if a<0). In vertex form, the turning point of the parabola is visible at a glance: (h,k). In standard form, you must use the formula x=-b/2a to find the vertex's x-coordinate, then solve for y. Vertex form is superior when the question asks for vertex, axis of symmetry, or transformations.

Example: y=2(x-3)²+5. Vertex is (3,5), opens upward (a=2>0), axis of symmetry is x=3.

Vertex to standard: Expand y=a(x-h)²+k. Example: y=2(x-3)²+5 → y=2(x²-6x+9)+5 → y=2x²-12x+18+5 → y=2x²-12x+23. Standard to vertex: Complete the square. Example: y=x²+4x+1 → y=(x²+4x+4-4)+1 → y=(x+2)²-4+1 → y=(x+2)²-3. Vertex: (-2,-3). Know both directions. If given standard form, convert to vertex to find the vertex. If given vertex form, expand to standard to find coefficients or y-intercept.

Quick check: In y=(x+2)²-3, vertex is (-2,-3), NOT (2,3). The x-coordinate's sign flips.

Conversion 1: y=3(x-2)²+7 (vertex form). Vertex: (2,7). Standard: y=3(x²-4x+4)+7=3x²-12x+12+7=3x²-12x+19. Conversion 2: y=x²-6x+5 (standard). Complete the square: y=(x²-6x+9-9)+5=(x-3)²-9+5=(x-3)²-4. Vertex: (3,-4). Conversion 3: y=-(x+1)²+8 (vertex form). Vertex: (-1,8). Standard: y=-(x²+2x+1)+8=-x²-2x-1+8=-x²-2x+7. Conversion 4: y=2x²+8x+10 (standard). Factor 2: y=2(x²+4x+5). Complete: y=2((x²+4x+4-4)+5)=2((x+2)²+1)=2(x+2)²+2. Vertex: (-2,2). Complete all four daily until conversion is fast and automatic.

Verify by plugging the vertex back into the equation; y should equal k.

Parabola form conversion appears in 2-3 ACT Math questions per section. Knowing both forms and when to use each gives you flexibility to solve any parabola question. A student who converts between forms quickly answers parabola questions in 1-2 minutes; one who relies only on one form struggles.

Master form conversion in one study session. By test day, switching between forms becomes automatic.