1003162302
Level:
C
Find all the values of the real parameter \( m \) such that \( f(x)=-2(x-m)^2+3 \) is an even function.
\( m=0 \)
\( m=3 \)
\( m=-3 \)
\( m\in(-\infty;\infty) \)
Quadratic functions
1003162301
Level:
C
Find all the values of the real parameter \( a \) such that \( f(x)=ax^2-2 \) is decreasing on \( (0;\infty) \).
\( a\in(-\infty;0) \)
\( a\in(0;\infty) \)
\( a\in[2;+\infty) \)
\( a\in(-\infty;2] \)
1003108312
Level:
B
The graph of the function \( f \) is a parabola, vertex of which is \( [6;0] \) and \( f(2)= 8 \) is given. Find the function \( f \).
\( f(x)=\frac12(x-6)^2 \)
\( f(x)=-\frac12(x-6)^2 \)
\( f(x)=\frac12(x+6)^2 \)
\( f(x)=\frac12x^2+6 \)
1003108311
Level:
B
The quadratic function \( f \) has the minimum at \( x=-2 \) and its graph passes through the points \( [0;13] \), \( [-1; 4] \). Find the function \( f \).
\( f(x)=3(x+2)^2+1 \)
\( f(x)=-\frac59(x-2)^2+9 \)
\( f(x)=\frac59(x-2)^2+9 \)
\( f(x)=3(x+2)^2-1 \)
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 \( f \).
\( f(x)=-4x^2-16x-16 \)
\( f(x)=-4x^2-16x+16 \)
\( f(x)=-\frac49x^2+\frac{16}9x-\frac{16}9 \)
\( f(x)=4x^2-16x+16 \)
1003108305
Level:
B
Find the quadratic function \( f \) whose graph is passing through the points \( [0;-9] \), \( [2;-3] \), \( [6;-3] \).
\( f(x)=-\frac12x^2+4x-9 \)
\( f(x)=-\frac12x^2+4x-3 \)
\( f(x)=x^2-8x-9 \)
\( f(x)=-2x^2+7x-9 \)