We investigate different types of functions and their graphs, including polynomial, exponential, logarithmic, and trigonometric functions. Understanding transformations and domain restrictions is essential for accurate graphing and interpretation.
Transformations include translations, reflections, dilations, and combinations of these. For example, the graph of y = f(x - 2) + 3 represents a translation of y = f(x) right by 2 units and up by 3 units.
State the transformation from y = x² to y = (x - 3)² + 2.
Shift 3 units right and 2 units up.
Sketch y = -2f(x).
Reflection in the x-axis and vertical stretch by a factor of 2.
We use index and logarithmic laws to simplify expressions and solve equations such as aˣ = b and logₐ(x) = y.
Solve for x: 3ˣ = 81
3ˣ = 3⁴ → x = 4
Simplify: log(1000) + log(0.1)
log(1000 × 0.1) = log(100) = 2
The derivative represents the instantaneous rate of change or gradient of a function. We differentiate polynomials, exponentials, and trigonometric functions using rules of differentiation.
Find dy/dx if y = 3x² - 4x + 5.
dy/dx = 6x - 4
Find the gradient of the tangent to y = x³ at x = 2.
dy/dx = 3x² → 3(2²) = 12
Integration is the reverse of differentiation and is used to calculate the area under curves. For example, ∫x² dx = (x³)/3 + C.
Find ∫(2x + 3) dx.
x² + 3x + C
Find the area under y = x between x = 0 and x = 3.
∫₀³ x dx = [x²/2]₀³ = 9/2 = 4.5
We model random processes using probability rules, including conditional probability and independent events. We also explore continuous probability distributions, including the normal distribution.
The probability of event A given event B has occurred is P(A|B) = P(A ∩ B) / P(B), provided P(B) ≠ 0.
If P(A) = 0.6, P(B) = 0.5, and P(A ∩ B) = 0.3, find P(A|B).
P(A|B) = 0.3 / 0.5 = 0.6
For a normal distribution with mean 0 and standard deviation 1, find P(Z < 0).
P(Z < 0) = 0.5
We analyse data using summary statistics and probability models. Methods includes random sampling, transformations of variables, and correlation.
The correlation coefficient (r) measures the strength and direction of a linear relationship between two variables. A least-squares regression line models this relationship.
If r = 0.92, describe the relationship between x and y.
Strong positive linear relationship.
Given the regression equation y = 2x + 5, find y when x = 4.
y = 2(4) + 5 = 13