Functions and Their Graphs (OLevel's Pure Math)

1. Polynomial Graphs

Cubic Graphs ($y=a{x}^3+b{x}^2+cx+d$)

Cubics generally have an "S" shape. The direction depends on the coefficient $a$:

• Positive $a$: Starts low (bottom left), ends high (top right).
• Negative $a$: Starts high (top left), ends low (bottom right).

To sketch a cubic, find the y-intercept (let $x=0$) and the roots by factorizing the expression (let $y=0$).

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2. Reciprocal Graphs

These graphs involve a variable in the denominator and feature asymptotes—lines that the curve approaches but never touches.

Function	Shape Name	Asymptotes
$y=\frac{k}{x}$	Rectangular Hyperbola	$x=0,y=0$
$y=\frac{k}{x^2}$	"Volcano" Shape	$x=0,y=0$

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3. Graph Transformations

Given a parent function $y=f\left(x\right)$, you can transform it using the following rules:

Transformation	Effect on Graph
$f\left(x\right)+a$	Translation by vector $\left(\genfrac{}{}{0ex}{}{0}{a}\right)$ (Vertical shift)
$f(x+a)$	Translation by vector $\left(\genfrac{}{}{0ex}{}{-a}{0}\right)$ (Horizontal shift - opposite direction!)
$af\left(x\right)$	Vertical stretch by scale factor $a$
$f\left(ax\right)$	Horizontal stretch by scale factor $\frac{1}{a}$
$-f\left(x\right)$	Reflection in the x-axis
$f(-x)$	Reflection in the y-axis

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4. Finding Asymptotes of Rational Functions

For a function like $y=\frac{ax+b}{cx+d}$:

• Vertical Asymptote: Set the denominator to zero and solve for $x$ ($cx+d=0$).
• Horizontal Asymptote: Look at the ratio of the leading coefficients as $x\to \infty$ ($y=\frac{a}{c}$).

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Worked Example: Transforming a Curve

Question: The curve $C$ has the equation $y=x^2$. Find the new equation after a translation of $\left(\genfrac{}{}{0ex}{}{3}{-2}\right)$.

Solution:

1. A horizontal shift of +3 means replacing $x$ with $(x-3)$.
2. A vertical shift of -2 means subtracting 2 from the whole function.
3. New Equation: $$y=(x-3{)}^2-2$$