Let $a$ and $b$ be real numbers where $0 < b < 7$. Let us consider the maximum value $M$ and the minimum value $m$ of the quadratic function
$$f ( x ) = x ^ { 2 } - 6 x + a$$
over the interval $b \leqq x \leqq 7$.
The function $f ( x )$ can be represented as
$$f ( x ) = ( x - \mathbf { A } ) ^ { 2 } + a - \mathbf { B } .$$
(1) For each of $\mathbf { C }$ ~ $\mathbf { G }$ in the following statements, choose the correct answer from among (0) ~ (9) below.
We are to find $M$ and $m$. There are two cases.
(i) When $0 < b \leqq \mathbf { C }$, then
$$M = \mathbf { D } , \quad m = \mathbf { E } .$$
(ii) When $\mathbf { C } < b < 7$, then
$$M = \mathbf { F } , \quad m = \mathbf { G } .$$
(0) 0
(1) 1
(2) 2
(3) 3
(4) $a - 6$
(5) $a + 7$ (6) $a + 8$ (7) $a - 9$ (8) $b ^ { 2 } - 6 b + a$ (9) $b ^ { 2 } + 6 b + a$
(2) In the case that $M = 13$ and $m = 1$, we have
$$a = \mathbf { H } , \quad b = \mathbf { I } .$$

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
$$a + b \leqq \frac { \mathbf { I } } { \mathbf { J } } .$$

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
$$b = \mathbf { H } , \quad c = \mathbf { I J } .$$

Consider the quadratic function
$$y = 3 x ^ { 2 } - 6 .$$
(1) Suppose that the graph obtained by a parallel translation of the graph of $y = 3 x ^ { 2 } - 6$ passes through the two points $( 1,5 )$ and $( 4,14 )$. The quadratic function of this graph is
$$y = \mathbf { A } x ^ { 2 } - \mathbf { B C } x + \mathbf { D E } .$$
This graph is the parallel translation of the graph of $y = 3 x ^ { 2 } - 6$ by $\mathbf{F}$ in the $x$-direction and by $\mathbf { G }$ in the $y$-direction.
(2) The quadratic function having the graph which is symmetric to the graph of $y = 3 x ^ { 2 } - 6$ with respect to the straight line $y = c$ is
$$y = - \mathbf { H } x ^ { 2 } + \mathbf { I } c + \mathbf { J } .$$
When the graphs of the two quadratic functions (1) and (2) have just one common point, it follows that $c = \mathbf { K }$, and the coordinates of the common point are ( $\mathbf { L } , \mathbf { M }$ ).

Consider the quadratic function
$$y = 3 x ^ { 2 } - 6 .$$
(1) Suppose that the graph obtained by a parallel translation of the graph of $y = 3 x ^ { 2 } - 6$ passes through the two points $( 1,5 )$ and $( 4,14 )$. The quadratic function of this graph is
$$y = \mathbf { A } x ^ { 2 } - \mathbf { BC } x + \mathbf { D E } .$$
This graph is the parallel translation of the graph of $y = 3 x ^ { 2 } - 6$ by $\mathbf{F}$ in the $x$-direction and by $\mathbf { G }$ in the $y$-direction.
(2) The quadratic function having the graph which is symmetric to the graph of $y = 3 x ^ { 2 } - 6$ with respect to the straight line $y = c$ is
$$y = - \mathbf { H } x ^ { 2 } + \mathbf { I } c + \mathbf { J } .$$
When the graphs of the two quadratic functions (1) and (2) have just one common point, it follows that $c = \mathbf { K }$, and the coordinates of the common point are ( $\mathbf { L } , \mathbf { M }$ ).

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) $\quad m$ 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 $\mathbf { L M }$.

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 $\mathbf { Z }$.

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) $\quad g ( 4 ) = f ( 4 )$.
(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.

Given that a real-coefficient quadratic polynomial function $f ( x )$ satisfies $f ( - 1 ) = k , f ( 1 ) = 9 k , f ( 3 ) = - 15 k$, where $k > 0$. Let the $x$-coordinate of the vertex of the graph of $y = f ( x )$ be $a$. Select the correct option.
(1) $a \leq - 1$
(2) $- 1 < a < 1$
(3) $a = 1$
(4) $1 < a < 3$
(5) $3 \leq a$

Given a vector $\vec { v } = ( - 2,3 )$ and two points $A$ and $B$ on the coordinate plane, where the $x$-coordinate and $y$-coordinate of point $A$, and the $x$-coordinate and $y$-coordinate of point $B$ all lie in the interval $[ 0,1 ]$, what is the maximum value of $| \vec { v } + \overrightarrow { A B } |$?
(1) $\sqrt { 13 }$
(2) $\sqrt { 17 }$
(3) $3 \sqrt { 2 }$
(4) 5
(5) $\sqrt { 2 } + \sqrt { 13 }$

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

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 parabolas $f ( x )$ and $g ( x )$ whose graphs are shown above intersect each other at their vertices.
Given this, what is the value of $\mathbf { g } ( \mathbf { 0 } )$?
A) 3
B) 4
C) 5
D) 6
E) 7