Functions, graphs and transformations

What you’ll learn

• Sketch polynomial and reciprocal graphs using key features.
• Describe translations, stretches and reflections of a graph.

Understand the idea

A useful sketch records intercepts, turning behaviour, asymptotes and the end behaviour. Factorised polynomials make the roots visible: a simple root usually crosses the axis, whereas an even repeated root touches it. The reciprocal curves $y=k/x$ and $y=k/x^2$ exclude $x=0$ and approach coordinate-axis asymptotes.

Changes outside $f$ act on output coordinates, so $af(x)$ multiplies heights by $a$ and $f(x)+b$ moves heights by $b$. Changes inside $f$ act on input coordinates inversely: $f(x+c)$ shifts left by $c$, and $f(kx)$ divides horizontal coordinates by $k$. A negative scale also reflects the graph.

An intersection of $y=f(x)$ and $y=g(x)$ satisfies $f(x)=g(x)$. Use algebra to locate exact intersections and a sketch to check how many solutions are reasonable.

Key formulae

$$y=f(x)+b:\ (x,y)\mapsto(x,y+b)$$

$$y=f(x+c):\ (x,y)\mapsto(x-c,y)$$

$$y=af(kx):\ (x,y)\mapsto\left(\frac{x}{k},ay\right),\quad k\ne0$$

Worked example

The point $(2,-3)$ lies on $y=f(x)$. Find its corresponding point on $y=f(x-4)$.

1. The input changes from $x$ to $x-4$.
2. To keep the original input $2$, solve $x-4=2$.
3. Thus the new horizontal coordinate is $6$ and the height remains $-3$.

Answer: The corresponding point is $(6,-3)$, a translation 4 units right.