Quadratic functions

2000004302

Level: 
B
In the picture A, the graph of the quadratic function \( f(x) = x^2\) is shown. With the aid of the graph of \(f\) identify which of the graphs shown in the picture B is the graph of \( g(x) = -\frac{1}{2} x^2\). What is the color of the graph of \(g\)? (Note: Each graph in the picture B is the graph of some transformation of \(f\).)
green
blue
yellow
red

2000004301

Level: 
A
Find the intervals of monotonicity of the quadratic function \( f(x)=4-3x^2\).
The function is increasing on \( (-\infty; 0 ]\) and decreasing on \( [ 0 ; +\infty)\).
The function is increasing on \( (-\infty; 4 ]\) and decreasing on \( [ 4 ; +\infty)\).
The function is decreasing on \( (-\infty; 0 ]\) and increasing on \( [ 0 ; +\infty)\).
The function is decreasing on \( (-\infty; 4 ]\) and increasing on \( [ 4 ; +\infty)\).

1003124806

Level: 
C
We should fence the land in a shape of an equilateral triangle. Choose the function that describes the dependence of the fenced land area \( S \) (in square meters) on the length \( d \) (in meters) of the fence used.
\( S=\frac{\sqrt3}{36} d^2 \)
\( S=\frac{\sqrt3}{18} d^2 \)
\( S=\frac{\sqrt3}4 d^2 \)
\( S=\frac1{36} d^2 \)

1003124805

Level: 
C
On a spool of mass \( 0.5\,\mathrm{kg} \) is winded an aluminium wire of length \( 100\,\mathrm{m} \). Choose the function that describes the dependence of a mass of the spool with the wire \( m \) (in kilograms) on a diameter of the wire \( d \) (in millimetres). Wire density is \( 2\,700\frac{kg}{m^3} \). \[ \] Hint: The density of an object is defined as the ratio of the mass and the volume of the object.
\( m=\frac{27\pi}{400} d^2+0.5 \)
\( m= 67 500\pi d^2+0.5 \)
\( m=\frac{27\pi}{400} d^2-0.5 \)
\( m=\frac{27\pi}{200} d^2+0.5 \)

1003124804

Level: 
C
In the centre of a square shaped square there is a water fountain. The fountain has a square ground plan with the side length \( 4.5\,\mathrm{m} \). The square should be paved with cobblestones of size \( 25\,\mathrm{cm} \times 25\,\mathrm{cm} \). Choose the function that describes the dependence of the number of cobblestones needed (\( n \)) on the length of the square (\( a \)) given in meters.
\( n=16a^2-324 \)
\( n=\frac{a^2}{625}-324 \)
\( n=16a^2-625 \)
\( n=\frac{a^2}{16}-324 \)