Quadratic and Other Graphs

A quadratic function has the form $f(x)=a{x}^2+bx+c$ where $a\ne 0$. Its graph is a parabola.

Feature	Explanation
Shape	U-shape if $a>0$; ∩-shape if $a<0$
$y$-intercept	The value of $c$ (set $x=0$)
Roots	Where the graph crosses the $x$-axis: set $y=0$ and solve
Turning point	The minimum (if $a>0$) or maximum (if $a<0$) of the parabola
Line of symmetry	Vertical line through the turning point: $x=-\frac{b}{2a}$

Worked example — sketch $y=x^2-4x-5$:

• $y$-intercept: $c=-5$, so $(0,-5)$
• Roots: $x^2-4x-5=0⟹(x-5)(x+1)=0⟹x=5$ or $x=-1$
• Line of symmetry: $x=-\frac{-4}{2(1)}=2$
• Turning point: $x=2$; $y=4-8-5=-9$; minimum at $(2,-9)$ ✓

Completing the square rewrites a quadratic in the form $a(x+p{)}^2+q$, revealing the turning point at $(-p,q)$.

Worked example — find the turning point of $y=x^2-6x+11$:

$y=(x^2-6x)+11=(x-3{)}^2-9+11=(x-3{)}^2+2$

Turning point: $(3,2)$ — minimum because the coefficient of $x^2$ is positive ✓

Worked example — complete the square for $y=2x^2+8x-3$:

$y=2(x^2+4x)-3=2\left[(x+2{)}^2-4\right]-3=2(x+2{)}^2-8-3=2(x+2{)}^2-11$

Turning point: $(-2,-11)$ — minimum ✓

Cubic functions have the form $y=a{x}^3+b{x}^2+cx+d$.

• If $a>0$: goes from bottom-left to top-right; may have one or two "bends"
• If $a<0$: goes from top-left to bottom-right
• Simple example: $y=x^3$ passes through the origin, is symmetric about the origin

Key cubic sketch facts: $y=x^3-x=x(x-1)(x+1)$ has roots at $x=-1,0,1$.

Reciprocal function: $y=\frac{1}{x}$ (where $x\ne 0$)

• Two branches: one in the first quadrant ($x>0,y>0$) and one in the third ($x<0,y<0$)
• Has two asymptotes: the $x$-axis ($y=0$) and the $y$-axis ($x=0$) — the curve approaches but never touches them
• As $x\to \infty$, $y\to 0$; as $x\to 0^{+}$, $y\to +\infty$

(Extra context — the general reciprocal $y=\frac{k}{x}$ has the same shape, scaled by $k$; not required to be named explicitly.)

Exponential function: $y=k^x$ for a positive base $k\ne 1$

• If $k>1$: exponential growth — graph rises steeply for $x>0$, approaches zero for $x<0$
• If $0<k<1$: exponential decay — graph falls towards zero for $x>0$
• Key values: $k^0=1$ for any valid $k$, so every exponential graph passes through $(0,1)$

Trigonometric graphs (arguments in degrees):

Function	Period	Range	Key values
$y=\sin x$	360°	$[-1,1]$	$\sin 0=0$; $\sin 90°=1$; $\sin 180°=0$; $\sin 270°=-1$
$y=\cos x$	360°	$[-1,1]$	$\cos 0=1$; $\cos 90°=0$; $\cos 180°=-1$; $\cos 270°=0$
$y=\tan x$	180°	all reals	undefined at $90°,270°,\dots$; $\tan 0=0$; $\tan 45°=1$

The $\sin$ and $\cos$ graphs are identical in shape but shifted by 90°. $\tan$ has vertical asymptotes at $90°+180°n$.

Graphs of non-standard functions appear in problems involving real-world relationships where the rule connecting $x$ and $y$ is not a simple standard form.

Kinematic graphs:

• Distance-time graph: gradient = speed; horizontal section = stationary; slope down = returning towards start
• Speed-time graph (velocity-time): gradient = acceleration; area under graph = distance travelled

Worked example — a speed-time graph shows speed increasing linearly from 0 m/s to 20 m/s over 4 seconds, then constant at 20 m/s for 6 seconds.

Distance in first 4 s (triangle): $\frac{1}{2}\times 4\times 20=40$ m

Distance in next 6 s (rectangle): $6\times 20=120$ m

Total distance: $40+120=160$ m ✓

Approximate solutions from graphs: read off the $x$-values where the graph reaches a given $y$-value, or where two graphs intersect.

1. Quadratic roots — forgetting one root when factorised

$(x-3)(x+2)=0$ gives two roots: $x=3$ and $x=-2$. Both values make one factor zero and both are valid roots. Missing one root loses a mark.

2. Turning point coordinates — swapping $x$ and $y$

The turning point of $(x-3{)}^2+2$ is at $(3,2)$ — the number inside the bracket (with its sign changed) is the $x$-coordinate; the constant added is the $y$-coordinate.

3. Reciprocal graph — drawing a line through the origin

$y=\frac{1}{x}$ is not a line and does not pass through the origin ($x=0$ is undefined). The graph has two separate branches in the 1st and 3rd quadrants.

4. Exponential — starting at zero on the $y$-axis

$y=2^x$ passes through $(0,1)$ because $2^0=1$. The graph approaches zero but never reaches it as $x$ decreases. It does not cross the $x$-axis.

Mistake	Correction
"Turning point of $y=(x-4{)}^2+3$ is $(-4,3)$"	Turning point is $(4,3)$ — sign inside bracket reverses
"Roots of $y=x^2-9$ are $x=9$"	$x^2=9⟹x=\pm 3$ (two roots: 3 and $-3$)
"$y=3^x$ passes through $(0,0)$"	$3^0=1$, so it passes through $(0,1)$