Quadratic Graphs

Quadratic Graphs

The graph of any quadratic function \( y = ax^2 + bx + c \) is a smooth, symmetrical U-shaped (or ∩-shaped) curve called a parabola. Every parabola has a line of symmetry, an intercept with the y-axis, a turning point (vertex), and zero, one or two x-intercepts (roots).

Key Features

  • Shape: U-shape when \( a > 0 \) (minimum), ∩-shape when \( a < 0 \) (maximum).
  • Axis of symmetry: the vertical line \( x = -\dfrac{b}{2a} \).
  • Turning point (vertex): lies on the axis of symmetry. Substitute the x-value back into the equation to find y.
  • y-intercept: set \( x = 0 \); always equals \( c \).
  • Roots (x-intercepts): set \( y = 0 \) and solve \( ax^2 + bx + c = 0 \) by factorising, completing the square or the quadratic formula.
x y −1 0 1 2 3 0 1 2 −1 −2 −3 −4 x = 1 (−1, 0) (3, 0) (0, −3) (1, −4) min

Finding Roots Algebraically

Set \( y = 0 \) and solve. Factorising is fastest when the quadratic factorises neatly. Use the quadratic formula \( x = \dfrac{-b \pm \sqrt{b^2-4ac}}{2a} \) otherwise.

\[ y = x^2 - 2x - 3 = (x-3)(x+1) = 0 \implies x = 3 \text{ or } x = -1 \]

Number of Roots: the Discriminant

The discriminant \( \Delta = b^2 - 4ac \) determines how many times the parabola crosses the x-axis:

  • \( \Delta > 0 \): two distinct roots (graph crosses x-axis twice).
  • \( \Delta = 0 \): one repeated root (graph just touches x-axis at the vertex).
  • \( \Delta < 0 \): no real roots (graph does not cross the x-axis).

Worked Examples

For \( y = 2x^2 - 8x + 5 \), find the axis of symmetry, turning point and y-intercept.
\[ \text{Axis of symmetry: } x = -\frac{-8}{2(2)} = 2 \] \[ \text{Turning point: } y = 2(4)-8(2)+5 = 8-16+5 = -3 \quad \Rightarrow \quad (2,\ -3) \text{ (minimum)} \] \[ \text{y-intercept: } x=0 \Rightarrow y = 5 \quad \Rightarrow \quad (0, 5) \]
Find the roots of \( y = 3x^2 - 5x - 2 \) and sketch the graph.
\[ 3x^2 - 5x - 2 = (3x+1)(x-2) = 0 \implies x = -\frac{1}{3} \text{ or } x = 2 \]

y-intercept: \( c = -2 \). Axis of symmetry: \( x = \frac{5}{6} \). Turning point minimum between the two roots. Sketch: U-shape (\( a=3>0 \)) crossing x-axis at \(-\frac{1}{3}\) and \(2\).

How many real roots do these functions have? (i) \( y = x^2+4x+4 \) (ii) \( y = x^2+2x+5 \)

(i) \( \Delta = 16 - 16 = 0 \): one repeated root. The graph just touches the x-axis at \( x = -2 \).

(ii) \( \Delta = 4 - 20 = -16 < 0 \): no real roots. The graph is a U-shape that lies entirely above the x-axis.

 Key Takeaways

  • Parabola: U-shape when \( a > 0 \) (minimum), ∩-shape when \( a < 0 \) (maximum).
  • Axis of symmetry: \( x = -\frac{b}{2a} \). The turning point always lies on this line.
  • y-intercept: always \( (0, c) \). Roots: solve \( ax^2+bx+c=0 \) (factorise or formula).
  • Discriminant \( b^2-4ac \): positive → 2 roots, zero → 1 (repeated) root, negative → no real roots.
  • The turning point x-coordinate is the mean of the two roots (when real roots exist).