Vertex Form of a Quadratic
This topic explores the 'vertex form' of a quadratic equation, y = a(x + b)² + c, which directly reveals the graph's key features. Understanding how the parameters a, b, and c transform the basic y = x² parabola is crucial for quickly sketching graphs and identifying their turning points without a calculator.
Part of the ESAT Mathematics 2 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- The parameter 'c' causes a vertical translation. A positive 'c' shifts the parabola up by c units, and a negative 'c' shifts it down.
- The parameter 'b' causes a horizontal translation. A positive 'b' (in the form (x + b)²) shifts the parabola left by b units. A negative 'b' (e.g., (x - 3)²) shifts it right.
- The parameter 'a' controls the shape and orientation. If a > 0, the parabola opens upwards (a minimum point). If a < 0, it opens downwards (a maximum point).
- The magnitude of 'a' acts as a vertical stretch factor. If |a| > 1, the parabola becomes narrower. If 0 < |a| < 1, it becomes wider.
- The vertex (turning point) of the parabola is located at the coordinates (-b, c).
- The vertical line x = -b is the axis of symmetry for the parabola.
Diagram
Formulae
y = a(x + b)2 + c This is the vertex form of a quadratic. Use it to quickly identify the transformations applied to the base parabola y = x2.
Vertex = (-b, c) To find the coordinates of the turning point of a parabola directly from its vertex form.
Axis of Symmetry: x = -b To find the equation of the line of symmetry for a parabola given in vertex form.
Definitions
- Vertex Form
- The equation of a quadratic written as y = a(x + b)² + c. It is also known as the completed square form.
- Vertex
- The turning point of the parabola, which is either the minimum point (if opening upwards) or the maximum point (if opening downwards).
- Axis of Symmetry
- A vertical line that divides the parabola into two mirror-image halves. Its equation is x = -b for a quadratic in vertex form.
Worked example
A quadratic curve has the equation y = -3(x + 2)² + 10. Find the coordinates of its vertex and the point where it intersects the y-axis.
- 1
Compare the equation y = -3(x + 2)² + 10 to the general form y = a(x + b)² + c.
- 2
Identify the parameters:
a = -3, b = 2, and c = 10 - 3
The vertex is at the point (-b, c).
Substitute the values:
(-2, 10).
- 4
To find the y-intercept, set x = 0 in the equation.
- 5 Calculate y = -3(0 + 2)² + 10
- 6 y = -3(2)² + 10 = -3(4) + 10
- 7 y = -12 + 10 = -2
- 8
The y-intercept is at the point (0, -2).
Answer: The vertex is at (-2, 10) and the y-intercept is at (0, -2).
Common mistakes
- ×Mistaking the sign of the horizontal shift: (x + 2)² is a shift of 2 units to the LEFT, not the right. The x-coordinate of the vertex is -b.
- ×Incorrectly applying the order of operations when substituting values. In y = a(x + b)² + c, the bracket must be squared before multiplying by 'a'.
- ×Forgetting that a negative 'a' value inverts the parabola, changing a minimum point into a maximum point.
No-calculator tips
- ✓Instantly locate the vertex at (-b, c). Plot this point first when sketching.
- ✓Check the sign of 'a' immediately. If a > 0, sketch a U-shape. If a < 0, sketch an n-shape. This gives the graph's basic form.
- ✓To find the y-intercept quickly, just calculate a*b2 + c in your head. For y = -3(x + 2)² + 10, this is -3*(22) + 10 = -12 + 10 = -2.