Chapter 4

Functions

Functions

Chapter 4●Key points of study●

1The function y=ax

2⑴

☞P 81〜

When y is a function of x and the relationship between them is expressed as y=ax2 (a is constant and0), y is said to be proportional to the square of x.

aIn the function y=ax2, when you multiply the value of x by n, the corresponding value of y becomes n2⑵

times larger.

Chapter 42

Graphing the function y=ax2☞P 83〜⑴

The graph of function y=ax is a smooth curve called a parabola.

⑵

A parabola has an axis of symmetry (a.k.a., the axis of the parabola).

2axis

The point where the parabola and its axis intersect is called the vertex of
the parabola.
⑶

increasedecrease

When a>0,①

vertexthe graph opens upward.

② as the value of x increases, the value of y decreases within the range x≦0vertexand increases within the range x≧0.
③

y≧0 regardless of the value of x.

④

and when x=0, the value of y is 0, which is its minimum value.

⑷

increasedecrease

When a<0,①axis

the graph opens downward.

② as the value of x increases, the value of y increases within the range x≦0
and decreases within the range x≧0.
③

y≦0 regardless of the value of x.

④

and when x=0, the value of y is 0, which is its maximum value.

⑸

As the absolute value of a increases, the graph opens more narrowly. The graph of y=−ax2 is thereflection of the graph of y=ax2 about the x-axis.

3

Rate of change☞P 88〜（increase in y）
When y is a function of x, the value ofis called the rate of change.
（increase in x）⑴“In the function y=ax2, the rate of change as the value of x increases from p to q isaq2−ap2.”q−pThis is equal to the slope of the straight line that passes through two points, whose respective
x-coordinates are p and q, on the graph of function y=ax2.
⑵

480

The rate of change in the linear function y=mx+n is constant, while that in the function y=ax2 is not.

Point where the graphs of y=ax2 and y=mx+n intersect☞P 90〜①

Solve the quadratic equation ax2=mx+n to find the value of the x-coordinate.

②

Substitute the value of the x-coordinate into y=ax2 or y=mx+n to find the value of the y-coordinate.

Chapter 4Functions

Exercises

2Properties of the function  y=ax2

1 Answer the following questions.

Find the maximum and minimum values of y for each domain of x below for the function y=−4x2.

□⑴①−2≦x≦−1②③0≦x≦5−3≦x≦4

The range of y is b≦y≦48 when the domain of x is −5≦x≦8 for the function y=ax2. Find the□⑵values of a and b.

2 Answer the following questions.
□⑴

Find the rate of change as the value of x increases as follows for the function y=①

from 1 to 3□⑵

②

3 2x.
4

from −8 to −4

An object moving in a straight line travels y m in x seconds after it starts, and the relationshipbetween x and y can be expressed as y=4x2. Answer the following questions.
①

What is the objectʼs average speed (in meters per second) during the period between two andfour seconds after it starts to move?
② When the average speed between t and（t+2）seconds after it starts is 16 m per second, find thevalue of t.

3 In each figure below, find the coordinates of points A and B, where the parabola and the straight line
intersect, and then find the area of △ABC.

□⑴□⑵

4 Answer the following questions.
□⑴

The graphs of the function y=−2x2 and the linear function y=ax−6 intersect at points A and B.

When the x-coordinate of A is −1, find the value of a and the coordinates of B.
□⑵

The graphs of the function y=ax2 and the linear function y=bx+12 intersect at two points. Whenthe x-coordinates of the points of intersection are −3 and 6, find the values of a and b.

94

Chapter 7

The Pythagorean theorem

The Pythagorean theorem

Chapter 7●Key points of study●

1

The Pythagorean theorem and the converse of the Pythagorean theorem⑴☞P 167〜

When the lengths of two sides adjacent to the right angle in a right triangle are a and b
and the length of the hypotenuse is c, the relationship between the lengths of the threesides is a2+b2=c2. (the Pythagorean theorem)⑵

When the lengths of the three sides of a triangle are a, b, and c and the relationshipbetween them is a2+b2=c2, the triangle is a right triangle in which the length of thehypotenuse is c. (the converse of the Pythagorean theorem)

2

Ratio of three sides in a special triangle

3The Pythagorean theorem and plane figures

Chapter 7⑴Length of diagonals in a square☞P 171〜☞P 171〜⑵ Length of diagonals in a rectangle⑶

equilateral triangle

x= a2+b2

x= 2 aHeight and area of anh=⑷Distance between two points

Length of a chord in a circle¬=2 r −dThe Pythagorean theorem and solids☞P 180〜

4⑴2

Length of diagonals in a cuboid222

x= a +b +c166⑸AB= （x1−x2）+（y1−y2）22⑵⑹ Length of a tangent to a circle¬= d 2−r 2

2Height of a square pyramid
OH= OA −AH23
3 2a，S=a242⑶Height of a coneh= ¬ 2−r 2Chapter 7

The Pythagorean theorem

Let’s learn the basics
Problem

Proving that the Pythagorean theorem is true

2

There are four congruent right triangles in square ABCD on the right. Thelengths of the two sides adjacent to the right angle in each triangle are a and b andthe length of each respective hypotenuse is c. Using this figure, prove that thePythagorean theorem is true.
Answer

The total area of the four triangles is

1ab×4=2ab.
2

Quadrilateral EFGH is a square whose side length is a−b,
so its area is（a−b）2.
The total area of the four triangles and square EFGH is2
2ab+（a−b）=2ab+（a2−2ab+b2）=a2+b2. ……①

The side length of square ABCD is c, so its area is c2. ……②Since ① and ② are equal, it is true that a2+b2=c2.

3 There are two congruent right triangles in trapezoid ABCD on the right. The lengths□of the two sides adjacent to the right angle in each triangle are a and b and the lengthof each respective hypotenuse is c. When using this figure to prove that thePythagorean theorem is true, fill in the blanks below to complete the proof.
〔Proof〕 The area of trapezoid ABCD is expressed in terms of a and b as ⅰ .
The total area ofADE andECB is expressed in terms of a and b asⅱ .
ABE is expressed in terms of c as ⅲ .
1
Since ⅳ − ⅴ = c2,
222
it is true that a +b =c2.
The area of

4 In the figure on the right,

ABC is a right triangle in which ∠ACB=90°, and□each its side is shared by one of the following squares: ADEB, BFGC, ACIJ.
The straight line drawn from C intersects AB perpendicularly at point H andintersects DE at point K. Fill in the blanks below to complete the proof thatAC2+BC2=AB2.
Since CJ is a diagonal of a square, square ACIJ=2

ACJ. …①

Since they share the same base and have the same height,ACJ= ⅰ . …②

Since two pairs of sides and the angle between them are equal,
Since they share the same base and have the same height,

ABJ≡ ⅱ . …③

ADC= ⅲ . …④Since DH is a diagonal of a rectangle, rectangle ADKH=2 ⅳ . …⑤
It can be said from ① to ⑤ that square ACIJ=rectangle ⅴ . …⑥
Similarly, square BFGC=rectangle ⅵ . …⑦
It can be said from ⑥ and ⑦ that square ACIJ+square BFGC=square ⅶ .
Therefore, it is true that AC2+BC2=AB2.

168