Identify vertex, axis of symmetry, and roots from an equation or graph, and connect the algebra to the parabola's actual shape. The same information shows up in three different forms of the equation — knowing which form to read it from is half the skill. Every question comes with a written walkthrough of the form used.
In y = a(x−h)² + k, the vertex's x-coordinate is +h, not −h — the minus sign inside the parentheses flips the sign of the number that appears there.
x = −b/(2a) has a negative sign built in — plugging in a negative b and forgetting the formula's own negative sign cancels it out, which is easy to lose track of.
The vertex sits on the axis of symmetry, exactly halfway between the two roots — it isn't one of the roots itself unless the parabola touches the x-axis at exactly one point.
Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.
What is the vertex of y = 2(x − 3)² + 5?
A — (3, 5). In vertex form y = a(x−h)² + k, the vertex is (h, k) directly. Here h = 3 and k = 5, so the vertex is (3, 5). (-3, 5) (B) comes from reading the sign inside the parentheses literally instead of recognizing that (x−3) means h = 3, not h = −3.
Find the axis of symmetry of y = 3x² − 12x + 7.
A — x = 2. Use x = −b/(2a) with a=3 and b=−12: x = −(−12)/(2·3) = 12/6 = 2. x = -2 (B) comes from dropping the formula's built-in negative sign — computing −12/(2·3) directly instead of −(−12)/(2·3), losing track of the double negative.
A parabola has roots at x = −1 and x = 5, and passes through the point (2, −9). What is its equation in standard form?
A — y = x² − 4x − 5. Start from the factored form using the roots: y = a(x+1)(x−5). Substituting the point (2, −9): −9 = a(3)(−3) = −9a, so a = 1. Expanding (x+1)(x−5) gives x² − 4x − 5. y = x² + 4x − 5 (B) comes from a sign error building the factors from the roots — using (x−1) and (x+5) instead of (x+1) and (x−5), flipping which sign corresponds to each root.
Quadratic function problems are really about reading the right form — vertex form for the vertex, factored form for the roots, standard form for the y-intercept — instead of forcing one formula to answer everything.
Before calculating anything, name which form the equation is in and what you can read off directly. Untimed practice is where that recognition speeds up.
Move to timed sessions once form-recognition is instant. Converting between forms takes longer than reading one directly, so budget accordingly.
Pair quadratic functions with quadratic equations in a mock — finding the roots is the same skill from two different angles.
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