Practice Quadratic functions
A quadratic function has the form f(x) = ax² + bx + c with a nonzero a, and its graph is a parabola.
Its vertex, roots and opening describe its behaviour.
Exercises with solutions
A quadratic function has the form f(x) = ax² + bx + c with a ≠ 0. Its graph is a parabola. The vertex is its highest or lowest point and lies on the axis of symmetry x = −b ÷ (2a). If a > 0 the parabola opens upwards, if a < 0 downwards.
Exercise 1
Find the vertex of f(x) = (x − 2)² + 3.
Show solution and working
S(2|3)
In vertex form f(x) = (x − d)² + e you read the vertex directly: d = 2 and e = 3.
Exercise 2
In which direction does the parabola of f(x) = −2x² + 1 open?
Show solution and working
downwards
The factor in front of x² is −2 and therefore negative, so the parabola opens downwards.
Exercise 3
Find the zeros of f(x) = x² − 9.
Show solution and working
x = 3 and x = −3
Set f(x) = 0: x² = 9, so x = 3 or x = −3.
Exercise 4
How many zeros can a quadratic function have?
Show solution and working
none, one or two
The parabola can miss the x-axis entirely, touch it in exactly one point, or cross it in two points.
Common mistake
Confusing the vertex with the y-intercept. The y-intercept is c, while the vertex lies on the axis of symmetry. And overlooking the opening direction when a is negative.
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