Functions 11 · The Learning Library

Summary

This Grade 11 functions course builds the idea of a function and uses it to study quadratic, exponential, sinusoidal and discrete models. It is written for students in a university-preparation functions course and for the teachers and tutors who support them. Real contexts appear throughout, such as fencing problems, thrown objects, daylight hours and savings plans.

Chapters 1–2 lay the foundation. Introduction to Functions defines a function, domain and range, and the vertical-line test. It introduces $f(x)$ notation and the five parent functions $x$ , $x^2$ , $\sqrt{x}$ , $\frac{1}{x}$ and $|x|$ . Students find inverses of linear functions by reversing operations or by swapping $x$ and $y$ . They also see that the inverse's graph is the reflection of the original in $y=x$ . Equivalent Algebraic Expressions covers adding, subtracting and multiplying polynomials. It also covers factoring by sum-and-product, decomposition, difference of squares and grouping. The chapter introduces rational expressions and their restrictions, and warns that $(a+b)^2 \neq a^2+b^2$ .

Chapters 3–4 study two key function families. Quadratic Functions uses second differences to spot quadratics in tables. It compares standard, vertex form $a(x-h)^2+k$ and factored form $a(x-r)(x-s)$ . Students complete the square to find maximum or minimum values and model revenue and profit with $P(x)=R(x)-C(x)$ . They also find inverses of quadratics, which need a $\pm$ root and are not functions unless the domain is restricted. Exponential Functions extends exponents to zero, negative and rational values, such as $27^{2/3}=9$ . It defines $f(x)=ab^x$ and its key features: the asymptote $y=0$ , the intercept 1, and growth versus decay. The chapter shows how to tell linear, quadratic and exponential data apart and how the parameters $a$ , $k$ , $d$ and $c$ transform the graph.

Chapters 5–6 turn to trigonometry. Trigonometric Ratios introduces cosecant, secant and cotangent. Exact values for 30°, 45° and 60° come from special triangles. Students extend the ratios to angles up to 360° using principal and related acute angles, terminal-arm coordinates and the CAST rule, and they begin proving simple identities. Sinusoidal Functions defines periodic functions, period, axis and amplitude. It builds the graphs of $\sin\theta$ and $\cos\theta$ from rotation on a circle and analyzes $f(x)=a\sin(k(x-d))+c$ .

Chapters 7–8 treat discrete functions. Sequences and Series covers arithmetic and geometric sequences, the Fibonacci sequence, and Gauss's pairing method. It gives the sum formulas for arithmetic and geometric series and links Pascal's triangle to binomial expansion. Financial Applications compares simple interest (linear growth, arithmetic sequences) with compound interest (exponential growth, geometric sequences). It then covers present value and simple ordinary annuities.

Appendices A and B review prerequisite skills. These include exponent laws, the Pythagorean theorem, linear systems, solving quadratics, the sine and cosine laws, and graphing-calculator techniques. The calculator topics are zeros, maximum and minimum, intersections, regression, difference tables and the TVM Solver.

Key ideas:

A function pairs each input with exactly one output; its inverse swaps inputs and outputs, reflecting the graph in $y=x$ .
Equivalent expressions simplify to the same form; factoring and expanding are reverse processes.
Vertex form $f(x)=a(x-h)^2+k$ reveals the maximum or minimum; factored form reveals the zeros.
Constant first differences mean linear, constant second differences mean quadratic, and a constant ratio means exponential.
Rational exponents mean roots: $b^{m/n}=(\sqrt[n]{b})^m$ .
The CAST rule and related acute angles give trigonometric ratios for any angle from 0° to 360°.
In $a\sin(k(x-d))+c$ , $|a|$ is the amplitude, $k$ sets the period, $d$ shifts the graph sideways and $c$ sets the axis.
Compound interest $A=P(1+i)^n$ grows geometrically, while simple interest $A=P(1+rt)$ grows linearly.

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