Quadratic Equation – General Form

\(ax^{2}+bx+c=0\)

A quadratic equation can always be written as \(ax^{2}+bx+c=0\) with real numbers \(a, b, c\) and \(a\neq0\). In this form \(a\) is the quadratic coefficient, \(b\) is the linear coefficient, and \(c\) is the constant term.

The Parabola: Key Features

Labelled diagram of a parabola showing vertex, axis of symmetry, arms, and y-intercept on a grid

Diagram of a parabola with vertex, axis of symmetry and arms labeled

What to spot on the graph

The graph of any quadratic equation forms a U-shaped curve called a parabola.

Find the vertex at the top or bottom. The vertical line through it is the axis of symmetry. The arms pointing up or down show the opening direction.

Key Points:

  • Shape: U-shaped parabola
  • Vertex: highest or lowest point
  • Axis of symmetry: vertical line \(x = h\)
  • Opening direction: arms up if \(a > 0\), down if \(a < 0\)

Effect of Coefficient ‘a’ on Opening Direction

Positive \(a\)

\(a > 0\)
Parabola opens upward — looks like a smile.

Negative \(a\)

\(a < 0\)
Parabola opens downward — looks like a frown.

Key Similarity

Sign of \(a\) alone predicts whether the parabola opens up or down.

Coefficient ‘a’ and the Width of the Parabola

Overlay of three parabolas on same axes: y = 0.5x² (blue, wide), y = x² (green, normal), y = 2x² (purple, narrow)

Wide (blue), normal (green) and narrow (purple) parabolas

Effect of |a| on Width

The absolute value \(|a|\) alone decides how wide or steep a parabola is.

Larger \(|a|\) stretches the graph vertically, making it steeper and narrower.

Smaller \(|a|\) compresses it, producing a flatter, wider curve.

Key Points:

  • \(|a| > 1\): narrow, steep (vertical stretch)
  • \(|a| < 1\): wide, flat (vertical compression)
  • Width depends only on \(|a|\), not on the sign of \(a\).

Role of Coefficient ‘b’ – Moving the Axis

Two parabolas y = x² and y = x² + 4x + 3 shown on same grid, highlighting shifted vertex and axis

Parabolas before and after changing \(b\)

Horizontal shift & completing the square

Changing \(b\) shifts the entire parabola left or right.

The axis of symmetry is \(x=-\frac{b}{2a}\).

Completing the square confirms the vertex lies on this axis.

Key Points:

  • \(b>0\): axis moves left \(\frac{b}{2a}\) units.
  • \(b<0\): axis moves right \(\frac{|b|}{2a}\) units.
  • Shift leaves opening direction and width set by \(a\) unchanged.

Role of Coefficient ‘c’ – The y-Intercept

Series of parabolas y = x², y = x² + 2, y = x² – 3 stacked vertically to show intercept change

Same curve shifted up or down as \(c\) varies

Constant term = y-intercept

For \(y = ax^{2} + bx + c\), when \(x = 0\) we get \(y = c\).

So the parabola crosses the y-axis at \((0, c)\); changing \(c\) slides it vertically.

Key Points:

  • Constant term \(c\) equals the y-intercept.
  • \(c > 0\): graph shifts up  \(c < 0\): graph shifts down.
  • Shape and width stay unchanged—only vertical position moves.

Match Equation to Graph

Drag each quadratic equation to the parabola that matches its form. This reinforces form-to-graph recognition.

Draggable Items

\(\,y = x^{2}\, \)
\(\,y = (x-2)^{2} + 1\,\)
\(\,y = -x^{2}+4\,\)

Drop Zones

Graph A

Graph B

Graph C

Tip:

Check the vertex and whether the parabola opens up or down to spot the correct match.

Zeros, Roots & the Discriminant

\[\Delta = b^{2} - 4ac\]

Variable Definitions

a coefficient of \(x^{2}\)
b coefficient of \(x\)
c constant term
\(\Delta\) discriminant

Applications

\( \Delta > 0 \)

Two real roots — parabola crosses x-axis twice.

\( \Delta = 0 \)

One real root — parabola touches x-axis.

\( \Delta < 0 \)

No real roots — parabola misses x-axis.

Key Takeaways

Graphs of Quadratic Equations

Opening Direction

Sign of \(a\): \(a>0\) opens up, \(a<0\) opens down.

Parabola Width

Larger \(|a|\) → narrower curve; smaller \(|a|\) → wider curve.

Horizontal Position

Coefficient \(b\) moves vertex sideways; axis of symmetry \(x=-\frac{b}{2a}\).

Vertical Position

Constant \(c\) shifts graph up or down; gives the \(y\)-intercept.

Number of Roots

Discriminant \(b^{2}-4ac\): >0 two roots, =0 one root, <0 none.