Math · College algebra · Concept
Quadratic functions: vertex form and graphs
A quadratic function f(x) = ax² + bx + c graphs as a parabola that opens upward when a > 0 and downward when a < 0. Its turning point, the vertex, lies on the axis of symmetry x = −b/(2a), and the vertex form f(x) = a(x − h)² + k shows the vertex (h, k) directly. The vertex gives the function’s minimum or maximum value, which is why quadratics answer questions about the greatest height or the largest revenue.
Standard form and the shape
In f(x) = ax² + bx + c the sign of a sets the direction: a > 0 opens upward, with a minimum, and a < 0 opens downward, with a maximum. A larger |a| makes the parabola narrower. The constant c is the y-intercept, f(0).
The vertex and the axis of symmetry
The parabola is symmetric about the vertical line through its vertex, x = −b/(2a), which lies halfway between the x-intercepts when there are any. Substitute that x into f to get the y-coordinate of the vertex.
Vertex form
f(x) = a(x − h)² + k is the graph of y = ax² moved so that its vertex is at (h, k). Completing the square turns standard form into vertex form, and expanding turns it back.
Intercepts
The y-intercept is c. The x-intercepts are the solutions of f(x) = 0, found by factoring, completing the square or the quadratic formula. A parabola can cross the x-axis twice, touch it at its vertex, or miss it, as the sign of the discriminant b² − 4ac shows.
Maximum and minimum problems
When a quantity is a quadratic function of a variable, its greatest or least value comes at the vertex. A thrown ball’s height and a shop’s revenue as it changes a price are common examples: find the vertex, then check that the answer makes sense in the problem.
Common mistakes
- Dropping the minus sign in x = −b/(2a).
- Reading the vertex of a(x − h)² + k as (−h, k): f(x) = 2(x − 2)² − 5 has vertex (2, −5).
- Reporting the x-coordinate of the vertex as the maximum: the maximum value is the output, f(h).
- Forgetting to factor a out of the x-terms before completing the square.
Key terms
- Quadratic function
- A function f(x) = ax² + bx + c with a ≠ 0. Its graph is a parabola that opens upward when a > 0 and downward when a < 0.
- Vertex of a parabola
- The point where a parabola turns, on its axis of symmetry. For f(x) = ax² + bx + c it lies at x = −b/(2a), and its y-coordinate is the function’s minimum or maximum value.
- Axis of symmetry
- The vertical line x = −b/(2a) through the vertex of the parabola y = ax² + bx + c. Points of the parabola at equal distances on either side of it have equal heights.
- Vertex form
- The form f(x) = a(x − h)² + k of a quadratic function, which shows its vertex (h, k) directly. Completing the square turns standard form into vertex form.
- Parabola
- The set of points equally far from a fixed point, the focus, and a fixed line, the directrix. With its vertex at the origin, x² = 4py opens upward if p > 0, with focus (0, p) and directrix y = −p. The graph of every quadratic function is a parabola.
- Extremum
- A maximum or minimum value. A local extremum is highest or lowest among nearby values; an absolute extremum is highest or lowest over the whole domain, and it can sit at an endpoint where f′ isn’t zero.
- x-intercept
- A point where a graph crosses or touches the x-axis, so its y-value is 0. For y = f(x), its x-value is a zero (root) of f.
- y-intercept
- The point where a graph crosses the y-axis, found by setting x = 0: (0, f(0)) when 0 is in the domain.
Work through an example
For f(x) = 2x² − 8x + 3, find the axis of symmetry, the vertex, the minimum value, the vertex form and the intercepts.
Find the vertex and axis of a parabola →Sources and scope
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