Extended Algebra and graphs

Topics

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Sample objectives

What this unit covers

• Algebraic manipulation: 1. Simplify algebraic expressions by combining like terms, ensuring the result is in its simplest form, for example, transforming 2a + 3ab - 1 + 5a - 9ab + 4 into 7a - 6ab + 3. 2. Expand products of algebraic expressions, such as 3x(2x - 4y) and (3x + y)(x - 4y), including cases with more than two brackets like (x - 2)(x + 3)(2x + 1). 3. Factor expressions by taking out common factors, fully factorising them, for instance, converting 9x + 15xy into 3x(3x + 5y). 4. Factor expressions of specific forms, including: • ax + bx + kay + kby • ax - by • a + 2ab + b • ax + bx + c • ax + bx + cx. 5. Complete the square for expressions structured as ax² + bx + c.
• Algebraic fractions: Work with algebraic fractions by performing operations such as addition, subtraction, multiplication, and division. This includes simplifying and factorizing rational expressions, for example, manipulating expressions like (x/(x - 4)) + (3/(2x)) or (3a/(9a)) × (4/10).
• Indices II: Comprehend and apply indices, including positive, zero, negative, and fractional values. For example, solve equations like x • 32 = 2 x + 1 and x • 5 = 25. Additionally, grasp and utilize the rules governing indices, such as simplifying expressions like 2^3 - 4^1 • x or x^2 • x^3 • x^2 • x^(-2) • x^5. Note that knowledge of logarithms is not necessary.
• Equations: 1 Create expressions, equations, and formulas, such as expressing the product of two consecutive even numbers, including the formation of simultaneous equations. 2 Solve linear equations with one variable, for example: 3x + 4 = 10 and 5 - 2x = 3(x + 7). 3 Solve fractional equations that include numerical and algebraic denominators, such as x/2 = 4 and (2x + 1)/3 + x/2 = 1. 4 Solve simultaneous linear equations with two variables. 5 Solve simultaneous equations that include one linear and one non-linear equation, with powers no greater than two. 6 Solve quadratic equations through factorization, completing the square, and using the quadratic formula, with the expectation of providing solutions in surd form. The quadratic formula is provided in the List of formulas. 7 Rearrange formulas to make a specific variable the subject, including cases where the subject appears multiple times or involves powers or roots.
• Inequalities: Represent and interpret inequalities, including on a number line, using open circles for strict inequalities (<, >) and closed circles for inclusive inequalities (≤, ≥). Construct, solve, and interpret linear inequalities such as 3x < 2x + 4 and 3x - 2 < 7. Graphically represent and interpret linear inequalities in two variables, using broken lines for strict inequalities and solid lines for inclusive inequalities, with shading to indicate unwanted regions. List inequalities that define a specified region, excluding linear programming problems.
• Sequences: 1 Extend a given numerical sequence or pattern, potentially using subscript notation, such as T representing the nth term of sequence T. 2 Identify patterns within sequences, which may include linear, quadratic, cubic, and exponential term-to-term rules, as well as connections between various sequences and their simple combinations. 3 Determine and apply the nth term for sequences.
• Proportion: Understand and express both direct and inverse proportion using algebraic forms, including linear, square, square root, cube, and cube terms, to determine unknown quantities. Familiarity with the proportional symbol (∝) is necessary.
• Graphs in practical situations: 1. Utilize and understand graphs in real-world contexts, including estimating and interpreting the gradient of a tangent at a specific point, as well as working with travel and conversion graphs. 2. Create graphs based on provided data. 3. Use the concept of rate of change in basic kinematics, focusing on distance-time and speed-time graphs, including acceleration and deceleration. 4. Determine the distance traveled by calculating the area under a speed-time graph, considering only linear segments of the graph.
• Graphs of functions: Create value tables and draw, identify, and analyze graphs for functions such as y = x + x - 4, y = 2x + n (where n can be -2, -1, 0, 1, 2, 3), y = ax (with a being a rational number), and y = ab + c (where b is a positive integer). Additionally, solve related equations graphically, such as determining where a line intersects a curve and finding roots using graphical methods. Also, draw and interpret graphs that depict exponential growth and decay scenarios.
• Sketching curves: Identify, draw, and analyze graphs of various functions including linear, quadratic, cubic, reciprocal, and exponential functions. This includes understanding turning points, roots, and symmetry, as well as recognizing vertical and horizontal asymptotes. You should be able to find turning points of quadratic functions by completing the square.
• Differentiation: 1. Estimate the slopes of curves by drawing tangent lines. 2. Utilize the derivatives of functions in the form ax^n, where 'a' is a rational constant and 'n' is a non-negative integer, including simple combinations of up to three of these functions. 3. Use differentiation to find slopes and identify stationary points (turning points). 4. Differentiate between maximum and minimum points using methods such as: • creating an accurate sketch • applying the second derivative test • examining the gradient on either side of a turning point. Note that candidates are not required to find points of inflection.
• Functions: Understand the concept of functions, including their domain and range, and use function notation such as f(x) = 3x - 5, g(x) = 5/2, and h(x) = 2x + 3. Learn to determine inverse functions and create composite functions, for example, if f(x) = 3x + 5 and g(x) = x + 2, find fg(x) and gf(x) = g(f(x)). Provide answers as fractions in simplest form. Note that finding the domains and ranges of composite functions is not required. This topic may also involve mapping diagrams.