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Proportionality:
Definitions:
Direct proportion: Two quantities increase or decrease together → If 𝑦 ∝ 𝑥, then 𝑦 = 𝑘𝑥 for some constant 𝑘
Inverse proportion: One quantity increases while the other decreases → If 𝑦 ∝ 1/𝑥, then 𝑦 = 𝑘/𝑥
Examples
Direct: If y = 12 when x = 4: → k = 12 ÷ 4 = 3 → Rule: y = 3x
Inverse: If y = 5 when x = 2: → k = 5 × 2 = 10 → Rule: y = 10 ÷ x
Graphs:
Definitions:
Travel graph: Shows how distance changes over time
Conversion graph: Converts between units (e.g. currency, temperature)
Rate of change: How one quantity changes compared to another (e.g. speed = distance ÷ time)
Speed–time graph: Shows how speed changes over time
Distance–time graph: Shows how distance changes over time
Examples
Distance–Time Graph A flat horizontal line = stationary A steep line = fast movement A curved line = acceleration or deceleration
Speed–Time Graph Area under graph = distance Constant speed = horizontal line Sloping line = changing speed
Techniques
Table of values: Choose x-values, substitute into the function, calculate y-values
Plot points: Use a ruler for straight lines, smooth curves for quadratics/cubics
Graphical solutions:
Roots = where graph crosses x-axis
Intersections = where two graphs meet
Use graph to estimate solutions to equations
Example
Graph the function y = x² – 2x – 3
Table of values: Choose x = –1, 0, 1, 2, 3 Calculate y for each x
Plot points and draw a smooth curve
Find roots: where y = 0 → estimate x-values from graph
Linear
Quadratic
Cubic
Exponential
Differentiation:
Definitions:
Gradient of a curve: The slope at a specific point, found by drawing a tangent
Differentiation: A method to find the gradient function of a curve
Turning point: Where the gradient is zero (flat point on the graph. Can be seen in the quadratic graph above at coordinates (3,1))
Maximum point: A peak (curve turns down)
Minimum point: A dip (curve turns up)
Rules of Differentiation:
For any term of the form axⁿ:
d/dx (axⁿ) = a × n × xⁿ⁻¹
Examples:
d/dx (5x²) = 10x
d/dx (–3x³) = –9x²
d/dx (7) = 0 (constants disappear)
Applications
Find gradient at a point
Differentiate the function
Substitute x-value into the derivative. This gives you the exact slope at that point
Find turning points
Set derivative = 0
Solve for x
Substitute x into original function to find coordinates
Identify maxima/minima. Use one of:
Sketch and observe shape
Check gradient before and after turning point
Use second derivative (if given). If the value is positive, the curve is going upwards so it's a local minimum point. However, if it's negative, the curve is going downward so it's local maximum.
Functions:
Key Concepts
Function notation f(x) means “the value of function f when x is input”
Example:
If f(x) = 3x – 5, then f(2) = 3(2) – 5 = 1
Domain and range
Domain: Set of input values (x-values)
Range: Set of output values (y or f(x)-values)
Inverse functions
Notation: f⁻¹(x)
Reverses the effect of f(x)
To find:
Swap x and y
Solve for y to get f⁻¹(x)
Composite functions
Notation: gf(x) means “apply f first, then g”
Example: If f(x) = x + 2 and g(x) = 3x, then gf(x) = g(f(x)) = 3(x + 2) = 3x + 6
Examples
Function evaluation f(x) = 2x² + 3 → Find f(4) → f(4) = 2(4)² + 3 = 2(16) + 3 = 35
Inverse function f(x) = 3x – 5 → Let y = 3x – 5 → Solve: x = (y + 5)/3 → So f⁻¹(x) = (x + 5)/3
Composite function f(x) = x + 2, g(x) = (3x + 5)² → gf(x) = g(f(x)) = (3(x + 2) + 5)² = (3x + 6 + 5)² = (3x + 11)²
NOTES DONE BY FARIDA SABET
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