Cubic, exponential and logarithmic graphs

Cubic graphs

• A cubic has $x^3$ as its highest power:

$$y = ax^3 + bx^2 + cx + d$$

• The characteristic shape has up to two turning points and passes from one corner of the plane to the opposite one.

• The sign of $a$ decides the direction:

  • $a > 0$ — comes up from the bottom left, exits top right
  • $a < 0$ — comes down from the top left, exits bottom right

• A cubic always crosses the $x$-axis at least once, because it runs from $-\infty$ to $+\infty$ (or the reverse) and must pass through zero on the way.

Reading roots from factorised form

• Cubics are almost always given factorised, and that form hands you the roots:

$$y = a(x - p)(x - q)(x - r)$$

• The $x$-intercepts are $x = p$, $q$ and $r$.
• The $y$-intercept is found by setting $x = 0$: $y = a(-p)(-q)(-r)$.

Repeated roots change the shape

• The multiplicity of a root determines the behaviour at that point:

  • Single factor $(x - p)$ — the curve crosses the axis straight through
  • Squared factor $(x-p)^2$ — the curve touches and turns back, like a parabola
  • Cubed factor $(x-p)^3$ — the curve flattens and passes through with a horizontal tangent

• $y = (x-1)(x-3)^2$ crosses at $x = 1$ and touches at $x = 3$.

• This is a Merit-level distinction, and it lets you sketch accurately without any calculus.

Exponential graphs

• An exponential has the variable in the exponent:

$$y = a\,b^x$$

• Each part:

  • $a$ — the $y$-intercept, since $b^0 = 1$
  • $b$ — the base, which must be positive

• The base decides the behaviour:

  • $b > 1$ — growth, rising steeply to the right
  • $0 < b < 1$ — decay, falling towards the right

• Key features of every exponential:

  • A horizontal asymptote at $y = 0$ — the curve approaches the $x$-axis but never touches it
  • No $x$-intercept, for the same reason
  • It is always positive (for $a > 0$), so the range is $y > 0$
  • The domain is all real $x$

• Note that exponentials grow far faster than polynomials. $2^x$ overtakes $x^{10}$ eventually, however large the power looks.

Logarithmic graphs

• A logarithm is the inverse of an exponential:

$$y = \log_b x$$

• The graph is the reflection of $y = b^x$ in the line $y = x$, and every feature swaps over accordingly:

	Exponential $y = b^x$	Logarithm $y = \log_b x$
Domain	all real $x$	$x > 0$
Range	$y > 0$	all real $y$
Asymptote	horizontal, $y = 0$	vertical, $x = 0$
Intercept	$y$-intercept at 1	$x$-intercept at 1

• The domain restriction is the thing to remember. $\log$ of zero or a negative number does not exist, so the curve exists only to the right of the $y$-axis.
• It rises steeply at first, then flattens off — but it keeps rising forever, never reaching a ceiling.

Circles

• A circle centred on the origin with radius $r$:

$$x^2 + y^2 = r^2$$

• Centred at $(h, k)$:

$$(x - h)^2 + (y - k)^2 = r^2$$

• A circle is not a function — it fails the vertical line test, since most $x$-values give two $y$-values.

Absolute value

• $y = |x|$ is a V shape with its point at the origin.
• It takes the positive version of whatever is inside: $|{-3}| = 3$.
• $y = |x - 2| + 1$ moves the point to $(2, 1)$, using the same translation rules as any other graph.