Quadratic functions

1003083108

Level: 
C
The parabolas of the functions \( f \) and \( g \) have the same vertex \( V \) and \( f(x)=ax^2+c \), where \( a \) and \( c \) are nonzero real numbers. Find \( g(x) \) such that the graphs of \( f \) and \( g \) are symmetric about the vertex \( V \) and that \( y \)-axis is their line of symmetry.
\( g(x)=-ax^2+c\), i.e. the equations of \( f \) and \( g \) differ in the sign of the coefficient at the quadratic term only
\( g(x)=ax^2-c\), i.e. the equations of \( f \) and \( g \) differ in the sign of the coefficient at the linear term only
\( g(x)=-ax^2-c \), i.e. \( g(x)=-f(x) \)
None of the statements above is true.

1103083107

Level: 
B
The quadratic functions \( f \) and \( g \) that have the same vertex \( V \) are graphed in the picture. The graph of \( g \) is the reflection of the graph of \( f \) in the vertex \( V \). Also, both the graphs are symmetric across \( y \)-axis. Identify the true statement about \( f \) and \( g \).
The equations of \( f \) and \( g \) differ in the sign of the coefficient at the quadratic term only.
The equations of \( f \) and \( g \) differ in the sign of the coefficient at the linear term only.
The equations of \( f \) and \( g \) differ in the sign of the coefficient at the absolute term only.
None of the statements above is true.

1103082702

Level: 
C
Function \( f \) is given completely by the graph. Identify which of the following statements is true.
\( f(x)=\left|x^2-1\right|;\ x\in[-2;2] \)
\( f(x)=\left|x^2\right|-1;\ x\in[-2;2] \)
\( f(x)=-\left|x^2+1\right|;\ x\in[-2;2] \)
\( f(x)=\left|-x^2\right|+1;\ x\in[-2;2] \)

1103067809

Level: 
C
Given the graphs of the functions \( f(x)=\frac12x^2-3 \) and \( g(x)=\frac12x \), find the solution set of the following equation. \[ \left|\frac12 x^2-3\right|=\left|\frac12 x\right| \]
\( \{ -3; -2; 2; 3 \} \)
\( \{ -2; 3 \} \)
\( \{ 2; 3 \} \)
\( \left\{ -\sqrt6; -2; \sqrt6; 3 \right\} \)