Quadratic models on the SAT often ask for a maximum height, a minimum cost, or the input that produces it. Vertex form answers both instantly. The question is usually testing whether you can tell the x-coordinate of the vertex apart from the value of the function there, since both numbers appear among the choices.
In f of x equals a times the quantity x minus h squared plus k, the vertex is at h and k, and k is the minimum when a is positive and the maximum when a is negative.
Vertex form is f of x equals a times the quantity x minus h squared plus k. The vertex sits at the point with coordinates h and k. If a is positive the parabola opens upward, so k is the minimum value. If a is negative it opens downward, so k is the maximum. Nothing else in the form matters for this question type.
Take f of x equals 3 times the quantity x minus 2 squared plus 7. Since a is positive, this has a minimum. The minimum value is 7 and it occurs when x equals 2. If a question asks for the minimum value of the function, the answer is 7. If it asks for the value of x at which the minimum occurs, the answer is 2. Both 2 and 7 will be choices.
Signs again. In f of x equals the quantity x plus 5 squared minus 4, the vertex x-coordinate is negative 5, because x plus 5 is the same as x minus negative 5. In f of x equals negative 2 times the quantity x minus 1 squared plus 9, the negative a means the vertex is a maximum, so the largest output is 9 and it happens at x equals 1.
If the function arrives in standard form, you have two ways to the vertex. Complete the square, or use the fact that the x-coordinate of the vertex is negative b divided by 2a. For f of x equals x squared minus 10x plus 3, that gives x equals 10 over 2, which is 5. Then substitute back: f of 5 equals 25 minus 50 plus 3, which is negative 22, so the minimum value is negative 22.
Factored form gives a third path and is worth recognizing. The parabola is symmetric, so the vertex sits exactly halfway between the two zeros. If f of x equals the quantity x minus 2 times the quantity x minus 8, the zeros are 2 and 8, so the axis of symmetry is x equals 5, and substituting gives the vertex output. In context questions, this is often the quickest route to a maximum height or maximum revenue.
Answering with the x-coordinate of the vertex when the question asked for the minimum or maximum value of the function itself.
A population doubles every 3 years. If it is 1000 now, what will it be in 12 years?
A patient receives 500 milligrams of a drug, and each hour 8% of the drug still present is cleared. To the nearest milligram, how much remains after 6 hours?
Why: Each period multiplies by 1 - 8/100 = 0.92, so the amount is 500(0.92)^6 ≈ 303. The value 460 applies the decrease a single time, which leaves far too much. Answering 330 means you counted one period too few.
Use negative b divided by 2a to get the x-coordinate, then substitute that value back into the function to get the y-coordinate. It is faster than completing the square when you only need the vertex and not a rewritten equation.
Look at the sign of the coefficient on the squared term. Positive means the parabola opens upward and the vertex is the lowest point, so it is a minimum. Negative means it opens downward and the vertex is the highest point, so it is a maximum.