Graphs of Quadratic Equations
Loading simulation…
flagWhat you'll discover
- arrow_forwardDraw the graph of y = ax² + bx + c from a table of values
- arrow_forwardState the shape of the graph and how the sign of a decides it
- arrow_forwardFind the vertex and the axis of symmetry using x = −b/2a
- arrow_forwardUse the discriminant D = b² − 4ac to count the real roots
- arrow_forwardSolve a quadratic equation graphically and check against the factorised form
Shape and opening
The graph of y = ax² + bx + c (a ≠ 0) is a parabola. If a is positive it opens upwards and has a minimum; if a is negative it opens downwards and has a maximum. The bigger |a| is, the narrower the parabola.
The y-intercept is simply c, because putting x = 0 leaves y = c. That fixes one point without any calculation, and it is a standard first step in the answer.
Vertex, axis and roots
The parabola is symmetric about the vertical line x = −b/2a. That line is the axis of symmetry, and it passes through the vertex. Substituting that x back into the equation gives the y-coordinate of the vertex.
The roots are the x-intercepts, where y = 0. Solving ax² + bx + c = 0 with the formula x = (−b ± √(b² − 4ac))/2a gives them. The sign of D = b² − 4ac decides the case: D > 0 gives two distinct real roots and the parabola cuts the x-axis twice; D = 0 gives one repeated root and the parabola touches the axis, its vertex sitting exactly on it; D < 0 gives no real root and the parabola never meets the axis.
Answering it for marks
A typical SEE question gives a quadratic, asks for a table of values and a graph, and then asks you to solve the equation from the graph. Take x values on both sides of the vertex, at least five of them. Plot the points accurately, join them with a smooth free-hand curve (never a ruler) and label the axis of symmetry and the vertex.
To solve graphically, read the x-values where the curve crosses the x-axis. If the question says "solve graphically", writing only the algebraic roots without the graph loses most of the marks — and the graph must actually pass through those values.