Ontario Grade 12 Advanced Functions (MHF4U) curriculum and lessons

The course map includes Exponential and Logarithmic Functions, Trigonometric Functions, Polynomial and Rational Functions, and 1 other section. Follow the complete section list above for the full study path.

درس تابع‌های پیشرفته پایه دوازدهم انتاریو (MHF4U) چه مباحثی دارد؟ از جمله مباحث این درس تابع‌های نمایی و لگاریتمی، تابع‌های مثلثاتی، تابع‌های چندجمله‌ای و گویا است. فهرست کامل بخش‌ها در بالای صفحه قرار دارد.

69 of 69 written lessons are currently published and free to read. Open a linked lesson in the section map above. Check the official Ontario curriculum for exact requirements.

درس‌ها و برنامهٔ درسی تابع‌های پیشرفته پایه دوازدهم انتاریو را از کجا پیدا کنم؟ اکنون 69 درس از 69 درس نوشتاری منتشر شده و خواندن آن‌ها رایگان است. درس‌های دارای پیوند را در نقشهٔ بخش‌های بالا باز کنید. برای شرایط دقیق به برنامهٔ درسی رسمی انتاریو مراجعه کنید.

An exponent tells how many times a base is used as a factor. A logarithm reverses that question: it asks which exponent produces a…

Angles can be measured in degrees or radians. A degree divides a full turn into 360 equal parts. A radian measures an angle by…

A curve or a list of numbers does not always reveal exactly how a function is defined. Its rule gives the clearest test. In this…

A function connects an input to an output. For example, a cyclist’s distance can depend on time. A rate of change describes how much…

A logarithm answers an exponent question. For example, asking for means asking, “What exponent on gives ?” Since , the answer is .…

An angle can be measured in degrees or radians. Degrees divide a full turn into 360 equal parts. Radians describe an angle by…

A polynomial can be written in more than one way. The expressions and describe the same polynomial, but they make different features…

A rate of change describes how an output changes as an input changes. For example, distance can change as time passes. The distance…

Logarithms, transformations, equations, and applications.

• A1.1 · Interpret and evaluate logarithms (published written lesson)
• A1.2 · Approximate logarithms in any base with technology (published written lesson)
• A1.3 · Connect logarithmic and exponential equations (published written lesson)
• A1.4 · Apply the laws of logarithms and exponents (published written lesson)
• A2.1 · Graph logarithmic functions and identify key features (published written lesson)
• A2.2 · Relate exponential and logarithmic functions as inverses (published written lesson)
• A2.3 · Transform logarithmic function graphs (published written lesson)
• A2.4 · Solve applications using exponential and logarithmic graphs (published written lesson)
• A3.1 · Recognize equivalent exponential and logarithmic expressions (published written lesson)
• A3.2 · Solve exponential equations using common bases or logarithms (published written lesson)
• A3.3 · Solve simple logarithmic equations (published written lesson)
• A3.4 · Solve real exponential and logarithmic applications (published written lesson)

Radians, reciprocal functions, identities, equations, and applications.

• B1.1 · Define radian measure and convert degrees and radians (published written lesson)
• B1.2 · Represent radian measures exactly and approximately (published written lesson)
• B1.3 · Evaluate primary and reciprocal ratios in radians (published written lesson)
• B1.4 · Determine exact ratios for special radian angles (published written lesson)
• B2.1 · Graph sine and cosine functions in radians (published written lesson)
• B2.2 · Graph the tangent function in radians (published written lesson)
• B2.3 · Graph reciprocal trigonometric functions and asymptotes (published written lesson)
• B2.4 · Interpret amplitude, period, and phase shift (published written lesson)
• B2.5 · Sketch transformed sine and cosine functions (published written lesson)
• B2.6 · Determine a sinusoidal equation from its graph (published written lesson)
• B2.7 · Solve sinusoidal applications in radians (published written lesson)
• B3.1 · Recognize equivalent trigonometric expressions (published written lesson)
• B3.2 · Use compound-angle formulas for exact values (published written lesson)
• B3.3 · Prove trigonometric identities (published written lesson)
• B3.4 · Solve linear and quadratic trigonometric equations (published written lesson)

Representations, factoring, equations, inequalities, and applications.

• C1.1 · Recognize polynomial expressions and functions (published written lesson)
• C1.2 · Compare polynomial representations (published written lesson)
• C1.3 · Identify polynomial graph features and end behaviour (published written lesson)
• C1.4 · Distinguish polynomial, sinusoidal, and exponential functions (published written lesson)
• C1.5 · Connect factored form with intercepts and sketches (published written lesson)
• C1.6 · Transform polynomial function graphs (published written lesson)
• C1.7 · Build polynomial equations from given conditions (published written lesson)
• C1.8 · Determine a polynomial family from zeros and a point (published written lesson)
• C1.9 · Classify functions as even, odd, or neither (published written lesson)
• C2.1 · Graph reciprocals of linear and quadratic functions (published written lesson)
• C2.2 · Analyze linear-over-linear rational functions (published written lesson)
• C2.3 · Sketch rational functions from key features (published written lesson)
• C3.1 · Apply the remainder and factor theorems (published written lesson)
• C3.2 · Factor polynomials up to degree four (published written lesson)
• C3.3 · Connect real roots with x-intercepts (published written lesson)
• C3.4 · Solve polynomial equations up to degree four (published written lesson)
• C3.5 · Connect rational-function roots with intercepts (published written lesson)
• C3.6 · Solve simple rational equations (published written lesson)
• C3.7 · Solve polynomial and rational applications (published written lesson)
• C4.1 · Relate equation solutions to inequality solutions (published written lesson)
• C4.2 · Solve polynomial and rational inequalities graphically (published written lesson)
• C4.3 · Solve factorable polynomial inequalities algebraically (published written lesson)

Rates of change, function combinations, composition, and modelling.

• D1.1 · Interpret rates of change in multiple representations (published written lesson)
• D1.2 · Distinguish zero, constant, and changing rates (published written lesson)
• D1.3 · Sketch graphs from verbal rate descriptions (published written lesson)
• D1.4 · Calculate and interpret average rate of change (published written lesson)
• D1.5 · Compare instantaneous and average rates of change (published written lesson)
• D1.6 · Approximate instantaneous rates numerically (published written lesson)
• D1.7 · Connect secant and tangent slopes with rates (published written lesson)
• D1.8 · Approximate tangent slopes using secants (published written lesson)
• D1.9 · Solve rate-of-change applications (published written lesson)
• D2.1 · Analyze sums, differences, products, and quotients of functions (published written lesson)
• D2.2 · Model applications with combined functions (published written lesson)
• D2.3 · Determine properties of combined general functions (published written lesson)
• D2.4 · Compose functions numerically and graphically (published written lesson)
• D2.5 · Compose functions algebraically and determine domains (published written lesson)
• D2.6 · Solve applications involving function composition (published written lesson)
• D2.7 · Connect inverse composition with the identity function (published written lesson)
• D2.8 · Express transformations as compositions (published written lesson)
• D3.1 · Compare major families of functions (published written lesson)
• D3.2 · Solve non-algebraic equations and inequalities (published written lesson)
• D3.3 · Model applications by reasoning with functions (published written lesson)

Solve log₂(x − 1) = 3.

Worked method: Rewrite in exponential form: x − 1 = 2³ = 8, so x = 9. The domain condition x > 1 is satisfied.

Describe the end behaviour of f(x) = −2x³ + x.

Worked method: The leading term −2x³ controls the ends. As x → ∞, f(x) → −∞; as x → −∞, f(x) → ∞.

Solve 5^(x + 1) = 125.

Find the exact value of sin(5π/6).