Quadratic functions

1003108310

Level: 
B
The graph of the quadratic function \( f \) has the vertex at the point \( [3; -1] \) and it passes through the point \( [ -1; 3] \). Find the function \( f \).
\( f(x)=\frac14x^2-\frac32x+\frac54 \)
\( f(x)=\frac14x^2+\frac32x+\frac54 \)
\( f(x)=-\frac14x^2+\frac32x-\frac{13}4 \)
\( f(x)=x^2+6x+8 \)

1003108309

Level: 
B
The graph of the quadratic function \( f \) intersects coordinate axes at the points \( [-3;0] \), \( [1;0] \), \( \left[0;\frac32\right] \). Find the function \( f \).
\( f(x)=-\frac12(x+1)^2+2 \)
\( f(x)=-\frac12(x+1)^2+\frac12 \)
\( f(x)=-\frac12(x-1)^2+2 \)
\( f(x)=\frac12(x-1)^2+2 \)

1003108308

Level: 
B
Which of the following information are insufficient to determine quadratic function uniquely?
two intersection points with the \( x \)-axis and \( x \)-coordinate of the vertex
two intersection points with the \( x \)-axis and \( y \)-coordinate of the vertex
two intersection points with the \( x \)-axis and any other point of the function
coordinates of the vertex and the intersection with the \( y \)-axis

1003108307

Level: 
C
Choose the triple of points, such that the graph of any of the functions \( f(x)=ax^2+c \), where \( a\in\mathbb{R}\setminus{0} \), \( c\in\mathbb{R} \), does not pass through all three points.
\( [-2;5] \), \( [2;1] \), \( [0;3] \)
\( [-2;5] \), \( [2;5] \), \( [0;3] \)
\( [-2;5] \), \( [2;5] \), \( [0;7] \)
\( [-2;5] \), \( [0;0] \), \( [1;1] \)

1003108306

Level: 
B
The \( x \)-axis is the tangent line of the graph of the quadratic function \( f \). The point of tangency has the coordinates \( [-2;0] \). Given \( f(-1)=-4 \), find the function \( f \).
\( f(x)=-4x^2-16x-16 \)
\( f(x)=-4x^2-16x+16 \)
\( f(x)=-\frac49x^2+\frac{16}9x-\frac{16}9 \)
\( f(x)=4x^2-16x+16 \)

1003108303

Level: 
B
Maximum value of the quadratic function \( f \) is \( 2 \). The graph of \( f \) intersects the \( x \)-axis at the points \( [-1;0] \) and \( [3;0] \). Find the function \( f \).
\( f(x)=-\frac12x^2+x+\frac32 \)
\( f(x)=x^2-2x+3 \)
\( f(x)=x^2-2x-3 \)
\( f(x)=-\frac12x^2-x+\frac32 \)

1003108302

Level: 
B
The graph of the quadratic function \( f \) is a parabola with the vertex \( [2;5] \). The parabola intersects the \( y \)-axis at the point \( [0;3] \). Find the function \( f \).
\( f(x)=-\frac12(x-2)^2+5 \)
\( f(x)=-\frac12(x+2)^2+5 \)
\( f(x)=-2(x-2)^2+5 \)
\( f(x)=-2(x+2)^2+5 \)