Question 174
A função $f(x) = x^2 - 4x + 3$ tem vértice no ponto
(A) $(2, -1)$ (B) $(2, 1)$ (C) $(-2, -1)$ (D) $(-2, 1)$ (E) $(1, 0)$

A função $f(x) = x^2 - 4x + 3$ tem vértice no ponto
(A) $(1, 0)$ (B) $(2, -1)$ (C) $(3, 0)$ (D) $(2, 1)$ (E) $(4, 3)$

The interior of a cup was generated by the rotation of a parabola around an axis $z$, as shown in the figure.
The real function that expresses the parabola, in the Cartesian plane of the figure, is given by the law $f(x) = \frac{3}{2}x^{2} - 6x + C$, where $C$ is the measure of the height of the liquid contained in the cup, in centimetres. It is known that the point $V$, in the figure, represents the vertex of the parabola, located on the $x$ axis.
Under these conditions, the height of the liquid contained in the cup, in centimetres, is
(A) 1. (B) 2. (C) 4. (D) 5. (E) 6.

The parabola $y = x^2 - 6x + 8$ has vertex at:
(A) $(2, 0)$
(B) $(3, -1)$
(C) $(3, 1)$
(D) $(4, 0)$
(E) $(6, 8)$

Let $a, b, c$ be real numbers such that $a = a^2 + b^2 + c^2$. What is the smallest possible value of $b$?
(A) 0
(B) $-1$
(C) $-\frac{1}{4}$
(D) $-\frac{1}{2}$.

Let $a$ be a constant. Consider the quadratic function in $x$
$$y = 2 x ^ { 2 } + a x + 3 .$$
Suppose that the vertex of the graph of (1) is in the first (upper right-hand) quadrant.
(1) The range of values which $a$ can take is
$$\mathbf { A B } \sqrt { \mathbf { C } } < a < \mathbf { D } ,$$
and the least integer $a$ satisfying this inequality is $\mathbf{EF}$.
(2) Let $a = \mathrm { EF }$ in (1). Let
$$y = 2 x ^ { 2 } + p x + q$$
be the equation of the graph which is obtained by translating the graph of (1) by $- \frac { 1 } { n }$ in the $x$-direction and by $\frac { 6 } { n ^ { 2 } }$ in the $y$-direction. Then
$$p = \frac { \mathbf { G } } { n } - \mathbf { H } , \quad q = \frac { \mathbf { I } } { n ^ { 2 } } - \frac { \mathbf { J } } { n } + \mathbf { K } .$$
(3) The total number of natural numbers $n$ for which $p$ in (2) is an integer is $\mathbf { L }$. Among these $n$, consider the ones such that the value of $q$ is also an integer. Then $\mathbf { M }$ is the value of the $n$ for which $q$ takes the minimum value $\mathbf { N }$.

Consider the quadratic function
$$y = - x ^ { 2 } - a x + 3 .$$
(1) If $a > 0$ and the maximum value of function (1) is 7 , then $a = \square$. In this case, the equation of the axis of symmetry of the graph is $x = \mathbf { B C }$, and the $x$-coordinates of the points of intersection of this graph and the $x$-axis are $\mathbf { D E } \pm \sqrt { \mathbf { F } }.$
(2) If the curve obtained by translating the graph of function (1) by 2 in the $x$-direction and by $-3$ in the $y$-direction passes through $( - 3 , - 5 )$, then $a =$ $\square$ G.

Consider the quadratic function
$$y = - x ^ { 2 } - a x + 3 .$$
(1) If $a > 0$ and the maximum value of function (1) is 7 , then $a = \square$. In this case, the equation of the axis of symmetry of the graph is $x = \mathbf { B C }$, and the $x$-coordinates of the points of intersection of this graph and the $x$-axis are $\mathbf { D E } \pm \sqrt { \mathbf { F } }.$
(2) If the curve obtained by translating the graph of function (1) by 2 in the $x$-direction and by $-3$ in the $y$-direction passes through $( - 3 , - 5 )$, then $a =$ $\square$ G.

Consider the quadratic function in $x$
$$y = - \frac { 1 } { 8 } x ^ { 2 } + a x + b .$$
When we denote the coordinates of the vertex of the graph of (1) by $( p , q )$, we have
$$p = \mathbf { A } a , \quad q = \mathbf { B } a ^ { 2 } + b .$$
(1) When the vertex ( $p , q$ ) is on the straight line $x + y = 1 , a$ and $b$ satisfy
$$b = \mathbf { C D } a ^ { 2 } - \mathbf { E E } a + \mathbf { F } ,$$
and so $8 a + b$ is maximized at $a = \mathbf { G }$, and the maximum value is $\mathbf { H }$.
(2) When the graph of (1) is tangent to the $x$-axis, the range of values of $a + b$ is
$$a + b \leqq \frac { \mathbf { I } } { \mathbf { J } }$$

Consider the quadratic function in $x$
$$y = - \frac { 1 } { 8 } x ^ { 2 } + a x + b .$$
When we denote the coordinates of the vertex of the graph of (1) by $( p , q )$, we have
$$p = \mathbf { A } a , \quad q = \mathbf { B } a ^ { 2 } + b .$$
(1) When the vertex ( $p , q$ ) is on the straight line $x + y = 1 , a$ and $b$ satisfy
$$b = \mathbf { C D } a ^ { 2 } - \mathbf { E E } a + \mathbf { F } ,$$
and so $8 a + b$ is maximized at $a = \mathbf { G }$, and the maximum value is $\mathbf { H }$.
(2) When the graph of (1) is tangent to the $x$-axis, the range of values of $a + b$ is
$$a + b \leqq \frac { \mathbf { I } } { \mathbf { J } } .$$

Let $f(x) = 3ax^{2} + (1 - a)$ be a real coefficient polynomial function, where $-\frac{1}{2} \leq a \leq 1$. On the coordinate plane, let $\Gamma$ be the region enclosed by $y = f(x)$ and the $x$-axis for $-1 \leq x \leq 1$.
Prove that when $-1 \leq x \leq 1$, $f(x) \geq 0$ always holds. (Non-multiple choice question, 4 points)