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 us consider the real numbers $x , y , t$ and $u$ satisfying the following four conditions:
$$\begin{aligned} & y \geqq | x | \\ & x + y = t \\ & x ^ { 2 } + y ^ { 2 } = 12 \\ & x ^ { 3 } + y ^ { 3 } = u \end{aligned}$$
We are to find the ranges of values which $t$ and $u$ can take.
(1) From (1) and (3), we see that the point $( x , y )$ is located on the arc which is a quadrant of the circle having its center at the origin and the radius $\mathbf { A }$. Moreover, the coordinates of the end points of this arc are
$$( \sqrt { \mathbf { C } } , \sqrt { \mathbf { CD } } ) \text { and } ( - \sqrt { \mathbf { C } } , \sqrt { \mathbf { D } } ) .$$
From this and (2), we also see that the range of values which $t$ can take is
$$\mathbf { E } \leqq t \leqq \mathbf { F } . \mathbf { G } .$$
(2) Next, from (2) and (3), we have
$$x y = \frac { \mathbf { H } } { \mathbf { I } } \left( t ^ { 2 } - \mathbf { J K } \right)$$
and further, using (4) we also have
$$u = \frac { \mathbf { L } } { \mathbf{L}} \left( \mathbf{NO} \, t - t ^ { 3 } \right)$$
Hence, since
$$\frac { d u } { d t } = \frac { \mathbf { P } } { \mathbf { Q } } \left( \mathbf { RS } - t ^ { 2 } \right)$$
the range of values which $u$ can take under the condition (5) is
$$\mathbf { TL } \leqq u \leqq \mathbf { UV } \sqrt { \mathbf{ W } } .$$

Consider the quadratic function in $x$
$$y = a x ^ { 2 } + b x + c .$$
The function (1) takes its maximum value 16 at $x = 1$, its graph intersects the $x$-axis at two points, and the length of the segment connecting those two points is 8. We are to find the values of $a$, $b$ and $c$.
From the conditions, (1) can be represented as
$$y = a ( x - \mathbf { A } ) ^ { 2 } + \mathbf { B } \mathbf { C }$$
and the coordinates of the two points at which the graph of (1) and the $x$-axis intersect are
$$( - \mathbf { D } , 0 ) , \quad ( \mathbf { E } , 0 ) .$$
Thus we obtain $a = \mathbf { F G }$. Hence we have
$$b = \mathbf { H } , \quad c = \mathbf { I J } .$$

Consider the quadratic function in $x$
$$y = a x ^ { 2 } + b x + c .$$
The function (1) takes its maximum value 16 at $x = 1$, its graph intersects the $x$-axis at two points, and the length of the segment connecting those two points is 8. We are to find the values of $a$, $b$ and $c$.
From the conditions, (1) can be represented as
$$y = a ( x - \mathbf { A } ) ^ { 2 } + \mathbf { B } \mathbf { C } ,$$
and the coordinates of the two points at which the graph of (1) and the $x$-axis intersect are
$$( - \mathbf { D } , 0 ) , \quad ( \mathbf { E } , 0 ) .$$
Thus we obtain $a = \mathbf { F G }$. Hence we have
$$b = \mathbf { H } , \quad c = \mathbf { I J } .$$

Q1 The quadratic function $f ( x ) = 2 x ^ { 2 } + a x - 1$ in $x$ satisfies
$$f ( - 1 ) \geqq - 3 , \quad f ( 2 ) \geqq 3 .$$
Let us consider the minimum value $m$ of $f ( x )$.
(1) $m$ can be expressed in terms of $a$ as
$$m = - \frac { \mathbf { A } } { \mathbf{B} } a ^ { 2 } - \mathbf { C }$$
(2) The range of the values of $a$ such that $f ( x )$ satisfies condition (1) is
$$\mathbf { D E } \leqq a \leqq \mathbf { F } .$$
(3) The value of $m$ is maximized when the axis of symmetry of the graph of $y = f ( x )$ is the straight line $x = \mathbf { G }$, and then the value of $m$ is $\mathbf { H I }$.
(4) The value of $m$ is minimized when the axis of symmetry of the graph of $y = f ( x )$ is the straight line $x = \mathbf { J K }$, and then the value of $m$ is $\mathbf { L M }$.

Let us consider the quadratic function
$$f ( x ) = \frac { 1 } { 4 } x ^ { 2 } - ( 2 a - 1 ) x + a ,$$
where $a$ is a real number.
(1) The coordinates of the vertex of the graph of $y = f ( x )$ are
$$\left( \mathbf { A } a - \mathbf { B } , - \mathbf { C } a ^ { 2 } + \mathbf { D } a - \mathbf { E } \right) .$$
(2) The range of $a$ such that the graph of $y = f ( x )$ and the $x$-axis intersect at two different points, A and B , is
$$a < \frac { \mathbf { F } } { \mathbf{G} } \text { or } \mathbf { H } < a .$$
(3) The range of $a$ such that the $x$-coordinates of both points A and B in (2) are greater than or equal to 0 and less than or equal to 6 is
$$\mathbf { I } < a \leqq \frac { \mathbf { J K } } { \mathbf { L M } } .$$

Let us consider the quadratic function
$$f ( x ) = \frac { 1 } { 4 } x ^ { 2 } - ( 2 a - 1 ) x + a$$
where $a$ is a real number.
(1) The coordinates of the vertex of the graph of $y = f ( x )$ are
$$\left( \mathbf { A } a - \mathbf { B } , - \mathbf { C } a ^ { 2 } + \mathbf { D } a - \mathbf { E } \right) .$$
(2) The range of $a$ such that the graph of $y = f ( x )$ and the $x$-axis intersect at two different points, A and B , is
$$a < \frac { \mathbf { F } } { \mathbf{G} } \quad \text { or } \mathbf { H } < a .$$
(3) The range of $a$ such that the $x$-coordinates of both points A and B in (2) are greater than or equal to 0 and less than or equal to 6 is
$$\mathbf { I } < a \leqq \frac { \mathbf { J K } } { \mathbf { L M } } .$$

Let us consider the maximum value $M$ and the minimum value $m$ of the quadratic function
$$f ( x ) = x ^ { 2 } - 2 ( a + 1 ) x + 2 a ^ { 2 }$$
over $0 \leqq x \leqq 2$, where $a$ is a constant and $0 \leqq a \leqq 3$.
(1) The coordinates of the vertex of the graph of $y = f ( x )$ are
$$\left( a + \mathbf { A } , a ^ { 2 } - \mathbf { B } a - \mathbf { C } \right) .$$
(2) For $\mathbf { D } \sim$ H in the following sentences, choose the correct answers from among choices (0) $\sim$ (9) below.
Let us find the maximum value $M$ and the minimum value $m$ according to the position of the axis of symmetry. We have that if $0 \leqq a < \mathbf { D }$, then
$$M = \mathbf { E } , \quad m = \mathbf { F } ;$$
if $\mathrm { D } \leqq a \leqq 3$, then
$$M = \mathbf { G } , \quad m = \mathbf { H } .$$
(0) 0
(1) 1
(2) 2
(3) 3
(4) $a ^ { 2 } - 2 a$
(5) $a ^ { 2 } - 2 a - 1$ (6) $2 a ^ { 2 }$ (7) $2 a ^ { 2 } - 2 a - 1$ (8) $2 a ^ { 2 } - 4 a$ (9) $2 a ^ { 2 } - 6 a + 3$
(3) Thus, $m$ is maximized at $a = \square$ and the value of $m$ then is $\square \mathbf { J }$. Also, $m$ is minimized at $a = \mathbf { K }$ and the value of $m$ then is $\mathbf { L M }$.

Let us consider the maximum value $M$ and the minimum value $m$ of the quadratic function
$$f ( x ) = x ^ { 2 } - 2 ( a + 1 ) x + 2 a ^ { 2 }$$
over $0 \leqq x \leqq 2$, where $a$ is a constant and $0 \leqq a \leqq 3$.
(1) The coordinates of the vertex of the graph of $y = f ( x )$ are
$$\left( a + \mathbf { A } , a ^ { 2 } - \mathbf { B } a - \mathbf { C } \right) .$$
(2) For $\mathbf { D } \sim$ H in the following sentences, choose the correct answers from among choices (0) $\sim$ (9) below.
Let us find the maximum value $M$ and the minimum value $m$ according to the position of the axis of symmetry. We have that if $0 \leqq a < \mathbf { D }$, then
$$M = \mathbf { E } , \quad m = \mathbf { F } ;$$
if $\mathrm { D } \leqq a \leqq 3$, then
$$M = \mathbf { G } , \quad m = \mathbf { H } .$$
(0) 0
(1) 1
(2) 2
(3) 3
(4) $a ^ { 2 } - 2 a$
(5) $a ^ { 2 } - 2 a - 1$ (6) $2 a ^ { 2 }$ (7) $2 a ^ { 2 } - 2 a - 1$ (8) $2 a ^ { 2 } - 4 a$ (9) $2 a ^ { 2 } - 6 a + 3$
(3) Thus, $m$ is maximized at $a = \square$ and the value of $m$ then is $\square$ J. Also, $m$ is minimized at $a = \mathbf { K }$ and the value of $m$ then is $\mathbf { L M }$.

Q1 For A $\sim$ K in the following sentences, choose the correct answer from among choices (0) $\sim$ (9) below. (1) Consider the quadratic function
$$y = a x ^ { 2 } + b x + c$$
whose graph is as shown in the figure at the right.
Then $a , b$ and $c$ satisfy the following expressions:
(i) $a \mathbf { A } 0 , b \mathbf { B } 0 , c \mathbf { C } 0$;
(ii) $a + b + c \mathbf { D } 0$;
(iii) $a - b + c \mathbf { E } 0$;
(iv) $4 a + 2 b + c \mathbf { F } 0$;
(v) $b ^ { 2 } - 4 a c \mathbf { G } 0$.
(2) Given the condition that $a , b$ and $c$ satisfy (i) and (ii) in (1), consider the case where the value of $a ^ { 2 } - 8 b - 8 c$ is minimized.
We see that $a = \mathbf { H }$. When we express $y = a x ^ { 2 } + b x + c$ in terms of $b$, we have
$$y = \mathbf { H } x ^ { 2 } + b x - b + \mathbf { I } \text {. }$$
Also, we see that the range of the values of $b$ is $\mathbf { J } < b < \mathbf { K }$. (0) 0
(1) 1
(2) 2
(3) 3
(4) 4
(5) - 2 (6) - 4 (7) $>$ (8) $=$ (9) $<$

(Course 2) Q1 For A $\sim$ K in the following sentences, choose the correct answer from among choices (0) $\sim$ (9) below. (1) Consider the quadratic function
$$y = a x ^ { 2 } + b x + c$$
whose graph is as shown in the figure at the right.
Then $a , b$ and $c$ satisfy the following expressions:
(i) $a \mathbf { A } 0 , b \mathbf { B } 0 , c \mathbf { C } 0$;
(ii) $a + b + c \mathbf { D } 0$;
(iii) $a - b + c \mathbf { E } 0$;
(iv) $4 a + 2 b + c \mathbf { F } 0$;
(v) $b ^ { 2 } - 4 a c \mathbf { G } 0$.
(2) Given the condition that $a , b$ and $c$ satisfy (i) and (ii) in (1), consider the case where the value of $a ^ { 2 } - 8 b - 8 c$ is minimized.
We see that $a = \mathbf { H }$. When we express $y = a x ^ { 2 } + b x + c$ in terms of $b$, we have
$$y = \mathbf { H } x ^ { 2 } + b x - b + \mathbf { I } \text {. }$$
Also, we see that the range of the values of $b$ is $\mathbf { J } < b < \mathbf { K }$. (0) 0
(1) 1
(2) 2
(3) 3
(4) 4
(5) - 2 (6) - 4 (7) $>$ (8) $=$ (9) $<$

Let $a$ be a positive constant. When we move the graph of the quadratic function $y = \frac{1}{4}x^2$ by parallel translation, the resulting parabola and the $x$-axis intersect at $(-2a, 0)$ and $(4a, 0)$. Let us consider the equation $y = f(x)$ of this parabola.
(1) The function $f(x)$ can be expressed as $$f(x) = \frac{\mathbf{A}}{\mathbf{B}}(x - \mathbf{C}a)(x + \mathbf{D}a)$$
(2) The range of values of $x$ such that the value of $y = f(x)$ is less than or equal to $10a^2$ can be obtained by solving the inequality $$x^2 - \mathbf{E}ax - \mathbf{FG}a^2 \leqq 0,$$ and it is $-\mathbf{H}a \leqq x \leqq \mathbf{I}a$.
(3) Suppose that the length of the segment between the intersections of the straight line $y = 10a$ and the graph of $y = f(x)$ is 10. Since $\mathbf{J}\sqrt{\mathbf{K}}a^2 + \mathbf{LM}a = 10$, we see that the value of $a$ is $\frac{\mathbf{N}}{\mathbf{O}}$.

Let $a$ be a positive constant. When we move the graph of the quadratic function $y = \frac{1}{4}x^2$ by parallel translation, the resulting parabola and the $x$-axis intersect at $(-2a, 0)$ and $(4a, 0)$. Let us consider the equation $y = f(x)$ of this parabola.
(1) The function $f(x)$ can be expressed as $$f(x) = \frac{\mathbf{A}}{\mathbf{B}}(x - \mathbf{C}a)(x + \mathbf{D}a)$$
(2) The range of values of $x$ such that the value of $y = f(x)$ is less than or equal to $10a^2$ can be obtained by solving the inequality $$x^2 - \mathbf{E}ax - \mathbf{FG}a^2 \leqq 0,$$ and it is $-\mathbf{H}a \leqq x \leqq \mathbf{I}a$.
(3) Suppose that the length of the segment between the intersections of the straight line $y = 10a$ and the graph of $y = f(x)$ is 10. Since $\mathbf{J}\sqrt{\mathbf{K}}a^2 + \mathbf{LM}a = 10$, we see that the value of $a$ is $\frac{\mathbf{N}}{\mathbf{O}}$.

Consider the two quadratic functions
$$f ( x ) = - 2 x ^ { 2 } , \quad g ( x ) = x ^ { 2 } + a x + b$$
Function $g ( x )$ satisfies the following two conditions:
(i) the value of $g ( x )$ is minimized at $x = 3$;
(ii) $g ( 4 ) = f ( 4 )$.
(1) From condition (i) we see that $a = -$ A . Further, from condition (ii) we see that $b = - \mathbf { B C }$. Hence the minimum value of function $g ( x )$ is $- \mathbf { D E }$.
(2) Let us find the value of $x$ such that $f ( x ) = g ( x )$ and $x$ is not 4 . Since $x$ satisfies
$$x ^ { 2 } - \mathbf { F } x - \mathbf { G } \mathbf { G } = 0 \text {, }$$
we obtain $x = - \mathbf { H }$.
(3) The value of $f ( x ) - g ( x )$ on $- \mathrm { H } \leqq x \leqq 4$ is maximized at $x = \square$, and its maximum value is JK.

The function $f ( x ) = x ^ { 2 } + a x + b$ satisfies the following two conditions:
(i) $\quad f ( 3 ) = 1$;
(ii) $13 \leqq f ( - 1 ) \leqq 25$.
We are to express the minimum value $m$ of $f ( x )$ in terms of $a$. In addition, we are to find the maximum and minimum values of $m$.
From condition (i), $a$ and $b$ satisfy
$$\mathbf { N } a + b + \mathbf { O } = 0 \text {. }$$
From this, $f ( x )$ can be expressed in terms of $a$ as
$$f ( x ) = x ^ { 2 } + a x - \mathbf { P } a - \mathbf { Q } .$$
Hence from condition (ii), $a$ satisfies
$$- \mathbf { R } \leqq a \leqq - \mathbf { S } .$$
On the other hand, $m$ can be expressed in terms of $a$ as
$$m = - \frac { 1 } { \mathbf { T } } ( a + \mathbf { U } ) ^ { 2 } + \mathbf { V }$$
Thus $m$ is maximized at $a = - \mathbf { W }$, and its maximum value is $\mathbf { X }$; it is minimized at $a = - \mathbf { Y }$, and its minimum value is $\mathbf { Z }$.

On the coordinate plane, there is a trapezoid with four vertices at $A ( 0,0 ) , B ( 1,0 ) , P , Q$ , where the line passing through $P$ and $Q$ has equation $y = 2 x + 4$ . If the coordinates of point $Q$ are $( a , 2 a + 4 )$ , where $a \geq 0$ is a real number, then the area of trapezoid $A B P Q$ is (14)$a +$ (16). (Reduce to lowest terms)

Let $f ( x ) , g ( x )$ both be real-coefficient polynomials, where $g ( x )$ is a quadratic with positive leading coefficient. It is known that the remainder when $( g ( x ) ) ^ { 2 }$ is divided by $f ( x )$ is $g ( x )$ , and the graph of $y = f ( x )$ has no intersection with the $x$-axis. Select the option that cannot be the $y$-coordinate of the vertex of the graph of $y = g ( x )$.
(1) $\frac { \sqrt { 2 } } { 2 }$
(2) 1
(3) $\sqrt { 2 }$
(4) 2
(5) $\pi$

Let the quadratic function $f ( x ) = x ^ { 2 } + b x + c$, where $b , c$ are real numbers. Given that $f ( x - 2 ) = f ( - x - 2 )$ holds for all real numbers $x$, and when $- 3 \leq x \leq 1$, the maximum value of $f ( x )$ is 4 times its minimum value, what is the minimum value of $f ( x )$?
(1) 0
(2) $\frac { 5 } { 3 }$
(3) 3
(4) 4
(5) 6

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 $B > b$ ?

$\mathrm { f } ( x )$ is a polynomial function defined for all real $x$.
Which of the following is a necessary condition for the inequality
$$\frac { \mathrm { f } ( a ) + \mathrm { f } ( b ) } { 2 } \geq \mathrm { f } \left( \frac { a + b } { 2 } \right)$$
to be true for all real numbers $a$ and $b$ with $a < b$ ?

$f(x)$ is a quadratic function in $x$. The graph of $y = f(x)$ passes through the point $(1, -1)$ and has a turning point at $(-1, 3)$.
Find an expression for $f(x)$.

The curve $S$ has equation
$$y = px^2 + 6x - q$$
where $p$ and $q$ are constants.
$S$ has a line of symmetry at $x = -\frac{1}{4}$ and touches the $x$-axis at exactly one point.
What is the value of $p + 8q$?
A $6$
B $18$
C $21$
D $25$
E $38$

A family of quadratic curves is given by
$$y _ { k } = 2 \left( x - \frac { k } { 2 } \right) ^ { 2 } + \frac { k ^ { 2 } } { 2 } + 4 k + 3$$
where $k$ is any real number and $y _ { k }$ is a function of $x$.
All these curves are sketched, and the point with the lowest $y$-coordinate among all the curves $y _ { k }$ is $( a , b )$.
Find the value of $a + b$

Here is an attempt to solve the inequality $x ^ { 4 } - 2 x ^ { 2 } - 3 < 0$ by completing the square:
$$x ^ { 4 } - 2 x ^ { 2 } - 3 < 0$$
I if and only if $x ^ { 4 } - 2 x ^ { 2 } + 1 < 4$ II if and only if $\left( x ^ { 2 } - 1 \right) ^ { 2 } < 4$ III if and only if $- 2 < x ^ { 2 } - 1 < 2$ IV if and only if $x ^ { 2 } - 1 < 2$ V if and only if $x ^ { 2 } < 3$ VI if and only if $- \sqrt { 3 } < x < \sqrt { 3 }$
Which of the following statements is true? A The argument is completely correct. B The first error occurs in line I. C The first error occurs in line II. D The first error occurs in line III. E The first error occurs in line IV. F The first error occurs in line V. G The first error occurs in line VI.

Consider the following multiple conditions on $x , y , z \in \mathbb { R }$.
$$\left\{ \begin{array} { c c c c c } 0 & < & z - x y & < & 1 \\ 0 & < & z - ( x + y ) ^ { 2 } & < & - x y \end{array} \right.$$
Let $\Omega$ be the set of points $( x , y )$ for which at least one $z$ exists satisfying the above conditions. Note that the set $\Omega$ can be seen in the three-dimensional Cartesian coordinate system as the orthogonal projection of points $( x , y , z )$ satisfying the above conditions onto the $x y$-plane. Answer the following questions.
(1) Find the inequalities on $x$ and $y$ representing $\Omega$.
(2) Draw a figure of $\Omega$ in the $x y$-plane. If the boundary of $\Omega$ intersects with the $x$-axis or the $y$-axis, write down the coordinates at each intersection.
(3) The curved segments of the boundary of $\Omega$ correspond to the linear transformation of arcs of the unit circle with a matrix $\mathbf { X }$. Find one such $\mathbf { X }$. Note that the point $( 1,0 )$ on the unit circle must be transformed to a point where the curvature is maximized in the curved segments.
(4) Calculate the determinant of $\mathbf { X }$ found in (3).
(5) Calculate the area of the set $\Omega$. Note that the absolute value of the determinant of a matrix is the area scale factor of the transformation with that matrix.