Functions · Ordinary Level
Functions
Ordinary Level · From your notes · Tap NEXT to begin
Section 5 of 6
Graphing Linear & Quadratic
Time to draw the functions. A linear rule gives a straight line; a quadratic gives a parabola. Once it’s on the page, every question becomes ‘look at the graph and read the answer’.
Lines & parabolas
Two points fix a line — work out $f$ at the ends of the domain and join with a ruler. For a parabola: $+x^2$ opens up (U-shape), $-x^2$ opens down (n-shape). Build a full table and draw a smooth curve.
Worked example — a straight line
Plot $f(x) = 3x + 1$ on $-2 \le x \le 3$.
Ends of the domain: $f(-2) = -5$, $f(3) = 10$
Plot the couples and join with a ruler
a straight line from $(-2,-5)$ to $(3,10)$
Worked example — a parabola
Plot $f(x) = x^2 - 2x - 3$ on $-2 \le x \le 4$.
Table: $-2\!:\!5,\ -1\!:\!0,\ 0\!:\!-3,\ 1\!:\!-4,\ 2\!:\!-3,\ 3\!:\!0,\ 4\!:\!5$
$+x^2$ → opens upwards. Draw a smooth U through the points
a U-shaped parabola, roots at $x=-1$ and $x=3$
Reading off a graph
$f(x) = $ a number → a horizontal line — where does it cut the curve? $f(x) = 0$ → the x-axis (the roots). The minimum / maximum is the very bottom / top of the curve.
Worked example — reading a parabola
Plot $f(x) = x^2 - 3x - 4$ on $-2 \le x \le 5$ and read off the roots and the minimum.
Roots (where it cuts the x-axis): $x = -1$ and $x = 4$
Minimum point: $(1.5,\ -6.2)$
roots $-1,\ 4$; minimum $(1.5,\ -6.2)$
You try
Plot $f(x) = -3x^2 + x + 1$ on $-1 \le x \le 1.5$ and find the maximum point.
$-x^2$ opens downwards, so it has a maximum at the top. Build the table and sketch the n-shape.
Table: $-1\!:\!-3,\ 0\!:\!1,\ 0.5\!:\!0.75,\ 1\!:\!-1,\ 1.5\!:\!-4.25$
The top of the curve is near $x \approx 0.2$, $y \approx 1.1$
a downward parabola with a maximum near $(0.2,\ 1.1)$
Worked example — the full reading question
Plot $f(x) = 2x^2 - 2x - 3$ on $-2 \le x \le 3$; state the roots and the minimum.
Roots (from the graph): about $x = -0.8$ and $x = 1.8$
Minimum point: $(0.5,\ -3.5)$
$f(x) < 0$ where the curve is below the x-axis: between the roots
roots $\approx -0.8,\ 1.8$; minimum $(0.5,\ -3.5)$
Part 5 done.
Next up: Cubic, Exponential & Applications. Head back to the hub for the next part.