Functions · Ordinary Level
Functions
Ordinary Level · From your notes · Tap NEXT to begin
Section 6 of 6
Cubic, Exponential & Applications
Bigger curves and real situations — but the method never changes: make a table, plot every point, draw a smooth curve, then read off.
The two new shapes
A cubic ($x^3$ term) makes an S-shape with two turning points — a local max and a local min. An exponential ($f(x)=2^x$) is always positive, always increasing, never crosses zero, and doubles every step.
Worked example — a cubic
Plot $f(x) = 2x^3 + x^2 - 13x + 6$ on $-3 \le x \le 2.5$; find the roots.
Roots (where it cuts the x-axis): $x = -3$, $x = \tfrac{1}{2}$, $x = 2$
It has a local maximum (top of the first hump) and a local minimum (bottom of the valley)
an S-curve with roots $-3,\ \tfrac{1}{2},\ 2$
Worked example — an exponential
Plot $f(x) = 2^x$ on $-1 \le x \le 4$; find $f(3)$ and where it doubles.
$f(3) = 2^3 = 8$
Every time $x$ goes up by $1$, the value doubles: $1, 2, 4, 8, 16$
$f(3)=8$; the curve doubles each step, never touching the x-axis
Real-world = same maths
Projectiles, areas and boxes are ordinary quadratics dressed up. Plot and read the answer — then say what it means physically.
Worked example — a projectile
A ball’s height after $x$ seconds is $h(x) = 35x - 5x^2$. Plot on $0 \le x \le 7$; find the maximum height.
Maximum is the top of the curve: $61.25$ m at $x = 3.5$ s
At $20$ m there are two times — once going up, once coming down
maximum height $61.25$ m at $t = 3.5$ s
You try
A rectangle has perimeter $14$ m; if its width is $x$, its area is $A(x) = 7x - x^2$. Plot on $0 \le x \le 7$ and find the maximum area.
It’s a downward parabola. The maximum is at the very top of the curve.
Table peaks at $x = 3.5$: $A(3.5) = 7(3.5) - 3.5^2 = 12.25$
Maximum area $12.25$ m$^2$ — when width = length (a square)
maximum area $12.25$ m$^2$ at $x = 3.5$ (a square)
That’s Functions, end to end.
Input-rule-output, domain & range, solving for $x$, composite functions, and graphing lines, parabolas, cubics, exponentials and real-world curves — every one from your notes.