Graphs of quadratic functions
The graph of a quadratic function is a parabola, a U-shaped curve with a single highest or lowest point. Its shape models projectiles, profits, and satellite dishes.
Parabolas
A quadratic function \(f(x) = ax^2 + bx + c\) graphs as a parabola. If \(a > 0\) it opens upward and has a minimum; if \(a < 0\) it opens downward and has a maximum. Larger \(|a|\) makes the parabola narrower; smaller \(|a|\) makes it wider.
Vertex and axis of symmetry
The vertex is the turning point. Its x-coordinate is \(x = -\frac{b}{2a}\), and the vertical line through it is the axis of symmetry, which splits the parabola into mirror images. For \(f(x) = x^2 - 6x + 5\), the vertex is at \(x = 3\), and \(f(3) = 9 - 18 + 5 = -4\), so the vertex is (3, -4).
Vertex form
Vertex form, \(f(x) = a(x - h)^2 + k\), shows the vertex (h, k) directly. Completing the square converts standard form: \(x^2 - 6x + 5 = (x^2 - 6x + 9) - 9 + 5 = (x - 3)^2 - 4\), confirming the vertex (3, -4).
Intercepts
The y-intercept is \(f(0) = c\). The x-intercepts are the real solutions of \(f(x) = 0\). For \(x^2 - 6x + 5 = (x - 1)(x - 5)\), they are x = 1 and x = 5, symmetric about the axis x = 3. A parabola can cross the x-axis twice, touch it once at its vertex, or miss it entirely.
Sketching a parabola
To sketch, find the direction of opening, the vertex, the axis of symmetry, the y-intercept and its mirror image across the axis, and any x-intercepts. These five or so points give an accurate shape without plotting a long table.
Maximum and minimum problems
The vertex gives the largest or smallest value of a quadratic. A farmer with 80 m of fencing for a rectangular pen against a wall, using three sides, gets area \(A = x(80 - 2x) = -2x^2 + 80x\). The maximum is at \(x = -\frac{80}{2(-2)} = 20\) m, giving \(A = 20 \times 40 = 800\) m².
Projectiles
Ignoring air resistance, the height of a projectile follows a quadratic in time. If \(h(t) = -4.9t^2 + 19.6t + 1.5\) metres, the peak is at \(t = \frac{19.6}{9.8} = 2\) s, with height \(h(2) = -19.6 + 39.2 + 1.5 = 21.1\) m. Parabolic shapes also appear in satellite dishes and car headlights, which focus signals and light.
Worked example
A theatre sells 300 tickets at $20 and estimates that each $1 price increase loses 10 sales. Revenue with x increases is \(R = (20 + x)(300 - 10x) = -10x^2 + 100x + 6000\). The vertex is at \(x = -\frac{100}{2(-10)} = 5\), so a $25 ticket gives maximum revenue: \(25 \times 250 = 6250\) dollars.
Key takeaways
Quadratic functions graph as parabolas that open up when a > 0 and down when a < 0, narrower as |a| grows. The vertex lies at x = -b/(2a) on the axis of symmetry. Vertex form a(x - h) squared + k shows the vertex (h, k) and comes from completing the square. The y-intercept is c, and x-intercepts are the real solutions of f(x) = 0. The vertex gives the maximum or minimum value, solving fencing, projectile, and revenue problems.
Flashcard
Graph of a quadratic function
Check your understanding
What is the vertex of \(f(x) = x^2 - 6x + 5\)?