Parabolas
Explore the graph of y = ax2 + bx + c, identify the vertex and axis of symmetry, find x-intercepts, and understand how a affects the shape.
The Basic Parabola y = x2
The graph of any quadratic function is a parabola. The simplest parabola is y = x2. It is symmetric about the y-axis and has its vertex (turning point) at the origin (0, 0).
The parabola y = x2 opens upward with vertex at (0, 0) and axis of symmetry x = 0.
a > 0
Opens upward
a < 0
Opens downward
Larger |a|
Narrower parabola
Key Features
For y = ax2 + bx + c, you need to identify the following features to sketch the parabola accurately.
Axis of symmetry
The vertical line x = −b/(2a). The parabola is a perfect mirror image on each side.
Vertex
Lies on the axis of symmetry. Find x = −b/(2a), then substitute to find y.
x-intercepts
Where y = 0. Set ax2 + bx + c = 0 and solve by factorising.
y-intercept
Substitute x = 0 to get y = c.
Finding the Vertex
Axis of Symmetry Formula
x = −b / (2a)
Example: Find the vertex of y = x2 − 4x + 1
a = 1, b = −4, c = 1
Step 1: Axis: x = −(−4)/(2×1) = 4/2 = 2
Step 2: y = 22 − 4(2) + 1 = 4 − 8 + 1 = −3
Vertex: (2, −3), Axis: x = 2
Key Vocabulary
Parabola
The U-shaped curve that is the graph of any quadratic function.
Vertex
The turning point; the minimum point when a > 0 or maximum when a < 0.
Axis of symmetry
The vertical line x = −b/(2a) through the vertex.
Concavity
Concave up (a > 0) means opening upward; concave down (a < 0) means opening downward.
Worked Examples
Sketch y = (x − 2)(x + 4)
x-intercepts: x = 2 or x = −4
Axis of symmetry: x = (2 + (−4))/2 = −1
Vertex: y = (−1 − 2)(−1 + 4) = (−3)(3) = −9. Vertex: (−1, −9)
y-intercept: y = (0 − 2)(0 + 4) = −8
Axis of symmetry of y = 2x2 + 8x − 3
a = 2, b = 8. Axis: x = −8/(2×2) = −2
y = 2(−2)2 + 8(−2) − 3 = 8 − 16 − 3 = −11
Vertex: (−2, −11)
Compare y = x2, y = 3x2, y = −x2
y = x2: opens upward, standard width.
y = 3x2: opens upward, narrower (|a| = 3 > 1).
y = −x2: opens downward, same width as y = x2.
Knowledge Check
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Key Concepts Summary
- ●The graph of a quadratic function is a parabola.
- ●If a > 0, it opens upward; if a < 0, it opens downward.
- ●The axis of symmetry is x = −b/(2a); the vertex lies on this line.
- ●The y-intercept is the value of c. Find x-intercepts by solving y = 0.
- ●A larger |a| produces a narrower parabola; a smaller |a| produces a wider one.