$$f(x|a,b,c) =
\begin{cases}
0 & \mathrm{for\ } x < a, \\
\frac{2(x-a)}{(b-a)(c-a)} & \mathrm{for\ } a \le x \leq c, \\[4pt]
\frac{2(b-x)}{(b-a)(b-c)} & \mathrm{for\ } c < x \le b, \\[4pt]
0 & \mathrm{for\ } b < x.
\end{cases}
\quad s.t. \quad
a \in (-\infty,\infty), \quad
a < b, \quad
a \le c\le b, \quad
a \le x \le b
$$

My childhood since the times of Kindergarten was full of simple scientific experiments. Here are some which I could remember:

Calculated the number of photo-receptors of my retina by using trigonometry after measuring my sight sharpness at a known distance. (13 yo)

Obtained a formula for angle between two planes if the line of their intersection, and a line connecting two non-intersection points of the planes are given. (16 yo)

Obtained and programmed a formula to obtain the distance between two given locations on Earth, assuming the Earth surface is spheric. (14 yo)

Discovered a formula to calculate the sum 1+2+3+...+n if n is given. (12 yo)

Soldered a solar panel from a set of solar cells, and used it for audio cassette player. (15 yo)

Made an automatic switch to control my air humidifier by a dry branch of fir that ascends or descends depending on the level of air humidity.

Disassembled a video projector from 1950s. (5 yo)

Read books, such as "ABC of Astronomy", "Science and Universe". (3 yo)

Planted cucumbers, water-melons and other plants, while trying to experiment what happens if I add various chemicals to the soil.

Collected chemical elements and their compounds. (14 yo)

Tried to extract Hypericin from St John's wort using chromatography in UV light.

Extracted oxygen from heated KMnO4.

Fused metals: Copper, Lead, Tin, Aluminum, etc.

Soldered...

Played with Electrolysis. Lacking batteries for a cassette player, tried to make them from electrolytes and electrodes.

Tried to extract N2O from NH4NO3 (but fearing its carcinogenic properties, didn't proceed).
(15 yo)

Bought and used a rotary evaporator for water distillation at home. (14 yo)

Tried to obtain ciano-compound by making a discharge between carbon electrodes (taken from batteries) in nitrogen atmosphere.

Made Iodine azide (an explosive compound) by putting crystals of Iodine into water solution of NH3. (14 yo)

Tried to capture Iodine from its ethanol solution by putting copper wires into the solution.

Tried to mix Aluminum powder, Magnesium slivers, KMnO4 and Sulfur powder, and ignite it.

Tried to make my fingerprint photo by dipping my fingers into aluminum powder, then sticking it to a transparent sticky sellotape, and sticking it to photo-film.

Made a solution of Luminol, which glows in the presence of complex Iron ions. (15 yo)

I tried to make a mirror by depositing silver from Silver nitrate onto glass. (14 yo)

I read a book "Numbers and Their Science" about natural numbers, number systems, prime numbers, Goldbach's conjecture. (12 yo)

I tried to make a concentrated NaCl solution, and tried to make a freezer by cooling the solution to -20 C, and circulating it through copper tubes...

When ill, I enjoyed reading books about Immunology and Genetics.

I tried to make stage smoke by heating glycerol on a pancake, and directing an air current onto its surface.

I tried to make St John's wort extract using izopropil alkohol.

I calculated the Centrifugal acceleration, and tried to make a centrifuge from an electric motor, then measured how many G`s can a cockroach withstand.

I made a terrarium, intended to keep a snake, but kept a little lizard, a turtle. I read books about herpetology.

I tried to inflate balloons with Hydrogen by putting Aluminum coins into water solution of NaOH.

I tried to mix glycerol with nitric acid in order to obtain nitroglycerin. (Very carefully)

I tried to make Hydrogen sulphide by putting ZnO into H2SO4.

I learned multiplication table of octal number system, and used it instead of decimal for making my calculations. (13 yo)