Quadratic Functions and Parabolas: f(x)=ax² and Transformations
A quadratic function is a function of the form f(x) = ax2 + bx + c with a ≠ 0, and its graph is a parabola. In brief: the simplest case is f(x) = x2, a U-shaped curve with its vertex at the origin. The sign of a decides whether the parabola opens up (a > 0) or down (a < 0), and |a| decides how narrow it is. For example, f(x) = 2x2 is narrower than x2, and f(x) = −(1/2)x2 is wider and opens downward.
Parabolas appear in the path of a thrown ball, in the shape of satellite dishes and in the arches of bridges. Understanding how the graph of f(x) = ax2 changes under shifts and reflections is the foundation for all work with quadratic functions. This guide explains the basic parabola, the effect of the coefficient a and the transformations that lead to the general form, with a value table and a figure.
What does the graph of f(x) = x² look like?
The graph of f(x) = x2 is the basic parabola. It passes through (0, 0), (1, 1), (−1, 1), (2, 4) and (−2, 4). It is symmetric about the y-axis, because (−x)2 = x2, which makes it an even function. The lowest point, the vertex, is the origin. The domain is all real numbers and the range is y ≥ 0.
The function decreases on (−∞, 0] and increases on [0, ∞). Unlike a V-shaped graph, the parabola is smooth at the vertex: it has a horizontal tangent there.
How does the coefficient a change the parabola?
- a > 0: the parabola opens upward and the vertex is the lowest point.
- a < 0: the parabola opens downward and the vertex is the highest point.
- |a| > 1: the parabola is narrower than y = x2 (vertical stretch).
- 0 < |a| < 1: the parabola is wider (vertical compression).
Example: for x = 3, x2 = 9, 2x2 = 18, (1/3)x2 = 3 and −x2 = −9. The coefficient multiplies every output.
How do shifts transform the parabola?
The graph of f(x) = a(x − h)2 + k is the graph of ax2 shifted h units to the right and k units up. Its vertex is (h, k) and its axis of symmetry is the line x = h.
Example 1: shift
f(x) = (x − 3)2 + 1 has vertex (3, 1), opens upward, and its y-intercept is f(0) = 9 + 1 = 10. It has no x-intercepts, since the minimum value is 1 > 0.
Example 2: reflection and stretch
f(x) = −2(x + 1)2 + 8 has vertex (−1, 8), opens downward and is narrower than x2. At x = 1: −2 · 4 + 8 = 0. At x = −3: −2 · 4 + 8 = 0. So the x-intercepts are 1 and −3.
Value table for f(x) = x² and f(x) = 2x²
| x | x2 | 2x2 | −x2/2 |
|---|---|---|---|
| −3 | 9 | 18 | −4.5 |
| −2 | 4 | 8 | −2 |
| −1 | 1 | 2 | −0.5 |
| 0 | 0 | 0 | 0 |
| 1 | 1 | 2 | −0.5 |
| 2 | 4 | 8 | −2 |
| 3 | 9 | 18 | −4.5 |
Each column is symmetric: the values at x and −x are equal. The pattern of differences 1, 3, 5, 7 for x2 (when stepping by 1) shows that the second differences are constant, which identifies a quadratic function in a table.
How do you graph a quadratic function step by step?
- Decide the direction from the sign of a.
- Find the vertex (h, k), and draw the axis of symmetry x = h.
- Compute the y-intercept f(0) and its mirror point across the axis.
- Find the x-intercepts if they exist.
- Add one or two more points and draw a smooth curve.
Example: f(x) = x2 − 4x + 3. In vertex form f(x) = (x − 2)2 − 1, so the vertex is (2, −1). The y-intercept is (0, 3) and its mirror point is (4, 3). The zeros come from (x − 1)(x − 3) = 0: x = 1 and x = 3. These points determine the parabola.
Where does the parabola come from?
The parabola was studied by Greek mathematicians as a conic section, a curve obtained by cutting a cone with a plane parallel to its side. Apollonius of Perga gave it the name “parabola” in his work on conics. Galileo showed in the early seventeenth century that a projectile moving under gravity follows a parabolic path, ignoring air resistance, which linked the curve to quadratic functions of time.
Where are parabolas used today?
A ball thrown with an initial upward speed of 20 m/s has height h(t) = 20t − 4.9t2 meters, which is a quadratic function. Satellite dishes and car headlights use the reflecting property of the parabola, which sends parallel rays to one focus. Suspension bridge cables hang in an almost parabolic shape, and costs and profits are often modelled with quadratic functions.
Which mistakes are common?
Connecting points with straight segments
A parabola is a smooth curve. Use enough points and draw a rounded vertex.
Shifting in the wrong direction
(x − 3)2 shifts the parabola to the right, and (x + 3)2 shifts it to the left.
Confusing the sign of a with the position of the vertex
The sign of a tells the opening direction. The vertex can lie above or below the x-axis in both cases.
Calling any curve with a bend a parabola
Only graphs of quadratic functions are parabolas. A cubic or a V-shape is a different curve.