How to find the rule of a step function from its graph

Read the graph in this order

Each feature of the picture fixes one parameter, and the order matters: $b$ comes first, because the sign of $a$ depends on it.

1. Which end of each step is closed? Closed at the left end: $b>0$. Closed at the right end: $b<0$.
2. How long is a step? Measure from one closed dot to the next. If that length is $L$, then $$\left|b\right|=\frac{1}{L}$$
3. How far apart are two neighbouring steps? That vertical distance is $\left|a\right|$.
4. Does the staircase climb or fall from left to right? If it climbs, $a$ has the same sign as $b$. If it falls, $a$ has the opposite sign.
5. Pick any closed dot and call it $(h,k)$.
6. Check. Evaluate your rule at the closed dot, at a point inside the same step, and at the step's open end, where the value must already belong to the next step.

Why a closed dot, and never an open one? Put $x=h$ into the rule: the bracket becomes $\lfloor 0\rfloor =0$, so $f(h)=k$. The point $(h,k)$ is on the graph, and since $b(x-h)$ passes the integer 0 there, it sits at the end of a step. A point of the graph at the end of a step is exactly what a closed dot is. An open dot is not on the graph at all, so it can never be $(h,k)$.

Example 1: a staircase that climbs

1. Closed ends. Every step is closed at its left end, so $b>0$.
2. Step length. The closed dots $(-2,1)$ and $(2,4)$ are 4 apart horizontally, so $\left|b\right|=\frac{1}{4}$ and $b=\frac{1}{4}$.
3. Jump. From height 1 to height 4 is 3, so $\left|a\right|=3$.
4. Sign of $a$. The staircase climbs and $b$ is positive, so $a=3$.
5. Closed dot. Take $(-2,1)$: $h=-2$ and $k=1$. Mind the sign: $x-h=x-(-2)=x+2$, a plus inside the bracket.
6. Rule. $$f(x)=3\lfloor \frac{1}{4}(x+2)\rfloor +1$$
7. Check. $f(-2)=3\lfloor 0\rfloor +1=1$, the closed dot. $f(1.9)=3\lfloor 0.975\rfloor +1=1$, still the same step. $f(2)=3\lfloor 1\rfloor +1=4$, so at $x=2$ the next step has begun and $(2,1)$ is rightly an open dot. $f(-6)=3\lfloor -1\rfloor +1=-2$, the height of the bottom step.

The rule is not unique

Starting from the other labelled closed dot, $(2,4)$, gives $$g(x)=3\lfloor \frac{1}{4}(x-2)\rfloor +4$$ which looks different but is the same function. For every $x$, $\frac{1}{4}(x+2)$ is exactly 1 more than $\frac{1}{4}(x-2)$, so its greatest integer is exactly 1 more too, and $3\times 1=3$ is precisely the difference between the two values of $k$. Any closed dot gives a correct rule, so if your answer differs from someone else's, test both at a few points before deciding that one of them is wrong.

Example 2: a staircase that falls

1. Closed ends. This time every step is closed at its right end, so $b<0$.
2. Step length. The closed dots $(1,3)$ and $(4,1)$ are 3 apart horizontally, so $\left|b\right|=\frac{1}{3}$ and $b=-\frac{1}{3}$.
3. Jump. From height 3 down to height 1 is 2, so $\left|a\right|=2$.
4. Sign of $a$. The staircase falls, so $a$ and $b$ have opposite signs. Since $b$ is negative, $a=2$, which is positive.
5. Closed dot. Take $(4,1)$: $h=4$ and $k=1$.
6. Rule. $$f(x)=2\lfloor -\frac{1}{3}(x-4)\rfloor +1$$
7. Check. $f(4)=2\lfloor 0\rfloor +1=1$, the closed dot. $f(1.3)=2\lfloor 0.9\rfloor +1=1$, still the same step. $f(1)=2\lfloor 1\rfloor +1=3$, so $x=1$ belongs to the higher step: the dot $(1,3)$ is closed and $(1,1)$ is open, as drawn. $f(4.3)=2\lfloor -0.1\rfloor +1=-1$, so just past 4 the next step down has begun.

Starting from the closed dot $(1,3)$ instead gives $2\lfloor -\frac{1}{3}(x-1)\rfloor +3$, the same function again.

All four cases at a glance

• Closed on the left, climbing: $b>0$ and $a>0$.
• Closed on the left, falling: $b>0$ and $a<0$.
• Closed on the right, climbing: $b<0$ and $a<0$. A price charged per started block has this shape.
• Closed on the right, falling: $b<0$ and $a>0$, as in Example 2.

Check your understanding

No calculator needed.

1. Every step of a staircase is 5 units long, and each one is closed at its left end. Which value of $b$ does that give?
2. True or false, with a reason: if every step of a graph is closed at its right end, then $a$ is negative.
3. A graph has three steps. Their closed dots are $(-1,2)$, $(1,6)$ and $(3,10)$, and their open dots are $(1,2)$, $(3,6)$ and $(5,10)$. Find a rule.
4. A step function equals $-2$ on $[0,4[$, $-5$ on $[4,8[$ and $-8$ on $[8,12[$, and the pattern continues in both directions. Find a rule, then find $f(-3)$.
5. A step function equals 7 on $]-3,-1]$, 4 on $]-1,1]$ and 1 on $]1,3]$. Find a rule.
6. A graph has the steps $[1,3[$ at height 0, $[3,5[$ at height 2 and $[5,7[$ at height 4. A student takes $(h,k)=(3,0)$ and writes $f(x)=2\lfloor \frac{1}{2}(x-3)\rfloor$. Find the error and correct the rule.
7. Do $f(x)=4\lfloor \frac{1}{3}x\rfloor +1$ and $g(x)=4\lfloor \frac{1}{3}(x-3)\rfloor +5$ have the same graph? Explain.
8. A staircase climbs by 2 at each jump, its steps are 0.5 units long and closed at their right ends, and $(1,0)$ is a closed dot. Find a rule, then find $f(0)$.

Practise it

To read parameters in both directions, from a rule to its staircase and back, work through the free Secondary 4 step functions worksheet. The Secondary 5 step functions worksheet opens with a section on finding the rule.

Practise finding the rule

• Secondary 4 Step Functions worksheet
• Secondary 5 Step Functions worksheet