Review Chapter 2-4 Chapter 2 Given that y is an integer, -3 < y ≤ 2.7, and y ≠ 0, write down all the possible values of y . It is given that z ≥ 41.5. Write down the least possible value of z if z is: an integer a prime number a perfect square Suggest a possible value of z for which z is a factor of 1260 and z is an even number. Solve the inequality 7 - (1/3) x ≥ 5 and represent the solution on a number line. A smartphone costs $750. Form an inequality and solve it to find the maximum number of smartphones that a company can buy with a budget of $25,000. Consecutive Even Numbers: Let the numbers be z , z +2, z +4. If the inequality z +( z +2)+( z +4)>120 is given, find the smallest even number z that satisfies this condition and calculate the product of t

Teori Trigonometri Trigonometri adalah cabang matematika yang mempelajari hubungan antara sisi dan sudut dalam segitiga. Berikut ini adalah penjelasan lengkap tentang konsep-konsep dasar trigonometri: Ukuran Sudut Terdapat dua jenis ukuran sudut yang umum digunakan dalam trigonometri: 1. Derajat : Satu putaran penuh = 360° $$1^{\circ} = \frac{1}{360} \text{ putaran}$$ 2. Radian : Perbandingan antara panjang busur dengan jari-jari lingkaran $$\alpha \text{ radian} = \frac{\text{panjang busur}}{\text{jari-jari}}$$ Hubungan antara derajat dan radian: $$1 \text{ radian} = \frac{180^{\circ}}{\pi}$$ $$1^{\circ} = \frac{\pi}{180} \text{ radian}$$ Fungsi Trigonometri Dasar Terdapat enam fungsi trigonometri dasar yang didefinisikan dalam segitiga siku-siku: 1. Sinus (sin) : $$\sin \alpha = \frac{\text{sisi depan}}{\text{sisi miring}}$$ 2. Cosinus (cos) : $$\cos \alpha = \frac{\text{sisi samping}}{\text{sisi miring}}$$ 3. Tangen (tan) : $$\tan \alpha = \frac{\text{sisi dep

Konsep Dasar dan Operasi Aljabar Panduan Lengkap untuk Pemahaman Aljabar 1. Pengertian dan Komponen Aljabar Komponen Utama Aljabar: Variabel (x, y, z) Koefisien (angka di depan variabel) Konstanta (angka tanpa variabel) Suku (term) Contoh Bentuk Aljabar: 3x² + 4x - 5 3x² 4x -5 2. Operasi Dasar Aljabar a. Penjumlahan dan Pengurangan Aturan Dasar: Hanya suku sejenis yang dapat dijumlahkan/dikurangkan Perhatikan tanda positif dan negatif Contoh: (5x² + 3x - 2) + (2x² - 4x + 1) = 7x² - x - 1 b. Perkalian Aljabar Rumus-rumus Penting: (a + b)² = a² + 2ab + b² (a - b)² = a² - 2ab + b² (a + b)(a - b) = a² - b² Visualisasi (a + b)² a² ab ab b² 3. Pemfaktoran Metode Pemfaktoran: Faktor Persekutuan Terbesar (FPB) 6x² + 12x = 6x(x + 2) Selisih Kuadrat x² - 25 = (x + 5)(x - 5) Kua

Surface Area, Volume, and Capacity: Comprehensive Guide 1. Surface Area Surface area is the total area of all faces or surfaces of a three-dimensional object. Common Shapes and Their Surface Area Formulas: Cube Surface Area = 6s², where s = length of side Rectangular Prism Surface Area = 2(lw + lh + wh), where: l = length w = width h = height Sphere Surface Area = 4πr², where r = radius Cylinder Surface Area = 2πr² + 2πrh, where: r = radius of base h = height 2. Volume Volume is the amount of three-dimensional space enclosed by a closed surface. Cube Volume = s³, where s = length of side Rectangular Prism Volume = l × w × h Sphere Volume = (4/3)πr³ Cylinder Volume = πr²h 3. Capacity Capacity refers to the maximum amount that something can contain, usually measured in units of volume. Common Conversions: 1 m³ = 1000 liters 1 liter = 1000 ml 1 cm³ = 1 ml 4. Practical Applica

Soal Transformasi Geometri Soal Transformasi Geometri 1. Tentukan persamaan bayangan dari fungsi linear \( f(x) = \frac{3}{2}x + \frac{5}{2} \) oleh pergeseran ke kiri sejauh 2 satuan dan ke atas sejauh 4 satuan. 2. Jika fungsi \( f(x) = 2x + 4 \) ditransformasikan terhadap translasi \( T = \begin{pmatrix} 2 \\ 5 \end{pmatrix} \), tentukan persamaan peta (bayangan) dari fungsi tersebut. 3. Sebuah fungsi linear \( f(x) = \frac{7}{5}x - 7 \) ditranslasi, sehingga bayangannya adalah \( g(x) = \frac{7}{5}x + 5 \). Tentukan matriks translasi fungsi tersebut. 4. Diketahui fungsi \( f(x) = x^2 + 4x - 12 \) ditranslasi oleh \( T = \begin{pmatrix} 3 \\ -5 \end{pmatrix} \). Tentukan persamaan peta dari fungsi tersebut. 5. Sebuah fungsi \( f(x) = -x^2 - 4x + 5 \) ditranslasi, sehingga bayangannya adalah \( g(x) = -x^2 - 8x - 8 \). Tentukan matriks translasi fungsi tersebut. 6. Parabola yang me

Aljabar: Konsep Dasar dan Operasi 1. Bentuk Aljabar Bentuk aljabar adalah ekspresi matematika yang terdiri dari variabel, konstanta, dan operasi matematika. Contoh: 3x + 2y - 7 Komponen Bentuk Aljabar: Variabel: Huruf yang mewakili nilai yang tidak diketahui (x, y, z) Koefisien: Angka yang mengalikan variabel (3 dalam 3x) Konstanta: Angka tanpa variabel (-7 dalam contoh di atas) Suku: Bagian dari bentuk aljabar yang dipisahkan oleh operasi penjumlahan atau pengurangan 2. Operasi Aljabar a. Penjumlahan dan Pengurangan Hanya suku-suku sejenis yang dapat dijumlahkan atau dikurangkan. Contoh: (3x + 2y) + (5x - 3y) = 8x - y b. Perkalian Gunakan sifat distributif untuk mengalikan bentuk aljabar. Contoh: (x + 3)(x + 7) = x² + 10x + 21 c. Pembagian Pembagian bentuk aljabar dapat dilakukan dengan memfaktorkan pembilang dan penyebut. Contoh: (x² - 4) ÷ (x - 2) = x + 2 3. Pemfaktoran Pemfaktoran adalah proses menguraikan bentuk aljabar menjadi

Comprehensive Guide to Trigonometry: A/AS Level Trigonometry is a fundamental branch of mathematics that deals with the relationships between the sides and angles of triangles. This comprehensive guide covers key concepts, formulas, and applications of trigonometry at the A/AS level. 1. Basic Trigonometric Ratios The three primary trigonometric ratios are sine, cosine, and tangent. These are defined in terms of the sides of a right-angled triangle: x (adjacent) y (opposite) r (hypotenuse) θ x = r cos θ y = r sin θ r² = x² + y² tan θ = y/x For an angle θ in a right-angled triangle: $$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{y}{r}$$ $$\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{x}{r}$$ $$\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{y}{x}$$ 2. Important Trigonometric Identities Several key identities are essential in trigono

Mastering Quadratic Functions: A Comprehensive Guide to Graphs and Properties Mastering Quadratic Functions: A Comprehensive Guide to Graphs and Properties Welcome to our in-depth exploration of quadratic functions and their graphs! Whether you're a student looking to ace your next math test or an enthusiast wanting to deepen your understanding, this guide will walk you through the essential concepts, properties, and applications of quadratic functions. 1. Introduction to Quadratic Functions A quadratic function is a polynomial function of degree 2, typically written in the form: f(x) = ax² + bx + c Where 'a', 'b', and 'c' are constants, and 'a' ≠ 0. The graph of a quadratic function is called a parabola, which has a distinctive U-shape (or inverted U-shape if 'a' is negative).

Sistem Koordinat Kartesius Sistem koordinat Kartesius merupakan metode untuk menentukan posisi objek pada bidang dua dimensi[1]. Sistem ini terdiri dari dua sumbu yang berpotongan tegak lurus: Sumbu X (horizontal) Sumbu Y (vertikal) Titik perpotongan kedua sumbu ini disebut titik asal (0,0)[1]. Komponen Utama Koordinat Kartesius Posisi pada Bidang Koordinat : Dinyatakan dalam bentuk pasangan terurut (x, y), di mana: x: jarak horizontal dari titik asal y: jarak vertikal dari titik asal Penentuan Letak Titik : Menggunakan titik asal (0,0) sebagai acuan, hitung langkah horizontal (x) dan vertikal (y)[1]. Jarak Terhadap Sumbu : Posisi objek dapat ditentukan berdasarkan jaraknya dari sumbu X dan Y[1]. Posisi Relatif : Lokasi titik dapat dinyatakan relatif terhadap titik asal atau titik acuan lain[1]. Posisi Garis : Garis dapat sejajar, tegak lurus, atau berpotongan dengan sumbu[1]. Kuadran dalam Koordinat Kartesius Bidang koor

Circle Theorems 1. Alternate Segment Theorem The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment. θ θ 2. Angle at the Centre Theorem The angle at the center is twice the angle at the circumference when both angles subtend the same arc. 2θ θ 3. Angles in the Same Segment Theorem Angles in the same segment are equal. θ θ 4. Angles in a Semicircle The angle in a semicircle is 90 degrees (a right angle). 90° 5. Chord of a Circle The perpendicular from the center of a circle to a chord bisects the chord (splits the chord into two equal parts). x x 6. Cyclic Quadrilateral The opposite angles in a cyclic quadrilateral are supplementary (they add up to 180°). a b a + b = 180° 7. Tangent of a Circle

Number Theory and Operations Understanding the relationships between different number representations is crucial in mathematics. This includes comparing fractions, decimals, and exponents. Ordering Numbers When ordering numbers in different forms, it's often helpful to convert them to a common representation. For example, when comparing fractions and decimals: Convert fractions to decimals: $$\frac{3}{5} = 0.6$$ Evaluate exponents: $$\left(\frac{3}{5}\right)^2 = 0.36$$ Irrational Numbers Some numbers, like $$\pi$$, are irrational. This means they cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal representations. Approximations like $$\frac{22}{7}$$ or 3.142 are often used in calculations, but they are not identical to the true value of $$\pi$$. Algebraic Expressions and Operations Evaluating Expressions When given an algebraic expression like $$p=-3^a-b^3$$, we can find its value by substituting known values for variables

Permutations and Combinations: A Comprehensive Summary 1. Permutations Permutations are arrangements of objects where order matters. Formula: For n distinct objects, the number of permutations is: P(n) = n! For r objects selected from n distinct objects: P(n,r) = n! / (n-r)! Example: Arranging the letters in "REQUIREMENT": 11! / (2!3!) = 3,326,400 (There are 11 letters, with 2 Rs and 3 Es repeated) 2. Combinations Combinations are selections of objects where order doesn't matter. Formula: Selecting r objects from n distinct objects: C(n,r) = n! / (r!(n-r)!) Example: Selecting 4 toys from 9 toys: C(9,4) = 9! / (4!5!) = 126 3. Conditional Probability in Selections When selecting objects with conditions, break down the problem into cases: Example: Selecting 6 people from 8 men and 4 women, with at least twice as many men as women: Case 1: 4 men, 2 women: C(8,4) * C(4,2) Case 2: 5 men, 1 woman: C(8,5) * C(4,1) Case 3: 6 men, 0 women: C(8,6) Total