Read and construct step graphs

Hours parked

\(\displaystyle 4\)

A car park charges $4 for stays from 0 hours up to 2 hours, $7 for stays from 2 hours up to 4 hours, and $12 for stays from 4 hours onwards, up to 8 hours. What is the charge for parking exactly 4 hours?

A locker charges $3 for 0 hours up to 3 hours, $8 for 3 hours up to 6 hours, and $14 for 6 hours onwards. Move h in half-hour steps from 2.5 to 3.5, then through 6, and watch the charge now and half an hour later

What did the charge do as h passed through 3?

Read a step graph

A step graph stays at one height across each band and jumps at a boundary. The filled dot shows which band owns the boundary value, so a boundary value is read, never averaged. In the examples below, b=4, c₁=7 and c₂=12.

1. 1Find the band Choose the interval that contains the input value \(\displaystyle h=4\Rightarrow h\ge4\)
2. 2Read the horizontal height Every band has one constant charge across its interval \(\displaystyle h\ge4\Rightarrow C=\$12\)
3. 3Use the endpoint dots A filled dot includes an endpoint and an open dot excludes it \(\displaystyle \bullet\ (4,\$12)\ \text{included},\ \circ\ (4,\$7)\ \text{excluded}\)
4. 4Keep the values constant Do not interpolate between two horizontal step heights \(\displaystyle C(4)=12,\ \text{not}\ \frac{7+12}{2}\)

Up to 500 g

\(\displaystyle \$9\)

Over 500 g up to 1 kg

\(\displaystyle \$13\)

Over 1 kg up to 3 kg

\(\displaystyle \$18\)

Over 3 kg up to 5 kg

\(\displaystyle \$25\)

Parcels are posted by mass. Up to 500 g costs $9, over 500 g up to 1 kg costs $13, over 1 kg up to 3 kg costs $18, and over 3 kg up to 5 kg costs $25. What is the cost to post a 1000 g parcel?

Check which band includes its right end: look at the filled and open dots at 1000 g.

Up to 500 g

\(\displaystyle \$9\)

Over 500 g up to 1 kg

\(\displaystyle \$13\)

Over 1 kg up to 3 kg

\(\displaystyle \$18\)

Over 3 kg up to 5 kg

\(\displaystyle \$25\)

Using the same postage rates, what is the total cost to post a 400 g parcel and a 2.6 kg parcel?

Convert 2.6 kg to grams, read the cost for each parcel from its band, then add the two costs

Car-park charge

\(\displaystyle C(h)=\begin{cases}4&0\le h<2\\7&2\le h<4\\12&h\ge4\end{cases}\)

Hours parked

\(\displaystyle 4\)

Which calculation gives the charge at exactly 4 hours?