Vertex and parameter conditions for a quadratic graph
The question requires finding parameter values by imposing conditions on the vertex of a parabola (e.g., vertex lies in a specific quadrant, on a given line, or yields a specified extremum), using the completed-square form to express vertex coordinates in terms of parameters.
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.
102-- The minimum value of the function $y = mx^2 - 12x + 5m - 1$ for $m = 2$ is the axis of symmetry of the parabola. What is $x$? (1) $x=2$ (2) $x=2.5$ (3) $x=3$ (4) $x=3.5$
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) $-\frac{1}{2}$.
17. Let $f ( x ) = ( 1 + b 2 ) x 2 + 2 b x + 1$ and let $m ( b )$ be the minimum value of $f ( x )$. As $b$ varies, the range of $m ( b )$ is: (A) $[ 0,1 ]$ (B) $[ 0,1 / 2 ]$ (C) $[ 1 / 2,1 ]$ (D) $[ 0,1 ]$
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)
The diagram below shows the graph of $y = x ^ { 2 } - 2 b x + c$. The vertex of this graph is at the point $P$. Which one of the following could be the graph of $y = x ^ { 2 } - 2 B x + c$, where $B > b$ ?