Graphs of functions and sketching curves
Graphs of functions
- Make a table of values, plot the coordinates, and join with a smooth curve.
- Roots are where the graph crosses the $x$-axis (the solutions of $y = 0$).
- The intersection of a line and a curve solves the two equations together.
Practice
The roots of a graph are where it:
Roots are the x-values where y = 0 — where the curve meets the x-axis.
Sketching curves
| Function | Shape |
|---|---|
| $y = mx + c$ | straight line (gradient $m$, intercept $c$) |
| $y = ax^2 + bx + c$ | a parabola (U if $a>0$) |
| $y = \tfrac{a}{x}$ | reciprocal (two curves, with asymptotes) |
- Completing the square gives the parabola's turning point: $y = (x+3)^2 - 8$ → $(-3, -8)$.
Practice
The graph of y = ax² + bx + c (with a > 0) is a:
A positive a gives a U-shaped parabola; a negative a gives an ∩ shape.
Practice
The curve y = (x + 3)² − 8 has its turning point at (−3, q). What is q?
Completed-square form (x+p)² + q has turning point (−p, q), so q = −8.
You've got it
Key idea
- roots = where the curve crosses the $x$-axis; intersection solves two equations
- a quadratic is a parabola (U if $a>0$, ∩ if $a<0$)
- the turning point of $(x+p)^2 + q$ is $(-p, q)$