Reading a graph
The graph of a function shows every point (x, y) that satisfies y = f(x). The y-intercept is where the curve crosses the y-axis (set x = 0); the x-intercepts, or roots, are where it crosses the x-axis (set y = 0).
Key idea
A straight line y = mx + c has gradient m and y-intercept c. A quadratic y = ax2 + bx + c is a parabola: it opens upward when a > 0 and downward when a < 0.
Common shapes
You should recognise these at a glance: a linear graph is a straight line; a quadratic is a U-shaped or n-shaped parabola; a cubic y = x3 passes through the origin with a characteristic S-shape; and a reciprocal y = 1/x is a hyperbola with the axes as asymptotes.
Worked example
Find the roots of y = x2 - 5x + 6.
Factorise: x2 - 5x + 6 = (x - 2)(x - 3). Setting y = 0 gives x = 2 or x = 3, so the curve crosses the x-axis at (2, 0) and (3, 0).
The y-intercept is found at x = 0: y = 6.
Turning points and intersections
A parabola has one turning point (a minimum if it opens up, a maximum if it opens down). To find where two graphs meet, solve their equations simultaneously; the solution gives the coordinates of the intersection.
Remember
- x = 0 gives the y-intercept; y = 0 gives the x-intercept(s).
- y = (x - a)2 has its vertex at (a, 0).
- The number of x-intercepts of a quadratic can be 0, 1 or 2.