📈 Mastering Curves & Crossings: Graphs of Cubic, Exponential & Reciprocal Functions!

Graphs are the visual language of Mathematics. By sketching and interpreting curves we can **see** how functions behave, spot turning points, and even solve equations without algebraic manipulation. For O Level success, confidently reading cubic, exponential and reciprocal graphs is a must-have skill!

Core Ideas to Remember

A function’s graph reveals where it is increasing, decreasing, or crossing an axis. Notice how **cubic curves** can change direction twice, how **exponential curves** rocket upwards or decay, and how **reciprocal curves** create twin hyperbola branches. Using a graph, we can quickly locate zeros (roots) or intersections, giving a **graphical solution** to equations.

Cubic: general form $y=ax^{3}+bx^{2}+cx+d$ – may have up to 3 real roots and 2 turning points.
Exponential: $y=ab^{x}$ – always positive, passes through $(0,a)$ , rapid growth if $b>1$ , decay if $0<b<1$ .
Reciprocal: $y=\dfrac{k}{x}$ or $y=\dfrac{k}{x^{2}}$ – asymptotes on the axes, never touches them.
Graphical solution: the $x$ -coordinates where two graphs meet give the solutions to their equation.

Worked Example: Graphical Roots of a Cubic

Problem: Draw the graph of

y = x^{3} - 4x

for

-3 \le x \le 3

and use it to solve

x^{3} - 4x = 0

Step 1: Create a table of values. For example, $x=-3,-2,-1,0,1,2,3$ gives $y=-9,8,3,0,-3,8,18$ respectively.
Step 2: Plot these points accurately on graph paper and join them with a smooth curve to reveal an S-shaped cubic.
Step 3: The $x$ -intercepts occur where the curve crosses the $x$ -axis ( $y=0$ ). From the graph you should see crossings at $x=-2,\;0,\;2$ .
Step 4: Therefore the graphical solutions are
$\boxed{x=-2,\;0,\;2}$
.

Keep practising by sketching different functions and finding intersections with lines such as

y=2

y=x

. The more curves you draw, the faster you’ll recognise patterns and ace graph questions in your exam!