Functions and graphs
Connect exponential and logarithmic models with graph features, numerical solutions and exact quadratic methods.
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Before you begin
Connect exponential and logarithmic models with graph features, numerical solutions and exact quadratic methods.
Learning goals
- Interpret a function as one output for each input
- Model and interpret exponential and logarithmic relationships
- Use logarithm laws with domain conditions
- Solve quadratic equations by suitable methods
- Formulate, use and interpret a mathematical model
- Use quadratic structure and discriminants to reason about roots
- Interpret and draw graphs of functions
- Solve simultaneous equations by justified methods
- Solve inequalities and represent their solution sets
Syllabus statements covered
- Include: concept of function as a rule or relationship where for every input there is only one output; functions e^x and ln x and their graphs; exponential growth and decay; logarithmic growth; equivalence of y = e^x and x = ln y; laws of logarithms; use of a graphing calculator or graphing software to graph a given function; characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes (horizontal and vertical). Exclude: problems involving finding domain and range; use of notation f: x ↦ x² + 5; change of base of logarithms.
- Include: conditions for a quadratic equation to have two real roots, two equal roots or no real roots; conditions for ax² + bx + c to be always positive or always negative; solving simultaneous equations, one linear and one quadratic, by substitution; solving quadratic equations and inequalities in one unknown analytically; solving inequalities by graphical methods; formulating an equation or a system of linear equations from a problem situation; finding the approximate solution of an equation or a system of linear equations using a graphing calculator or graphing software.
- Include: concept of function as a rule or relationship where for every input there is only one output; functions e^x and ln x and their graphs; exponential growth and decay; logarithmic growth; equivalence of y = e^x and x = ln y; laws of logarithms; use of a graphing calculator or graphing software to graph a given function; characteristics of graphs such as symmetry, intersections with the axes, turning points and asymptotes (horizontal and vertical). Exclude: problems involving finding domain and range; use of notation f: x ↦ x² + 5; change of base of logarithms.
- Include: conditions for a quadratic equation to have two real roots, two equal roots or no real roots; conditions for ax² + bx + c to be always positive or always negative; solving simultaneous equations, one linear and one quadratic, by substitution; solving quadratic equations and inequalities in one unknown analytically; solving inequalities by graphical methods; formulating an equation or a system of linear equations from a problem situation; finding the approximate solution of an equation or a system of linear equations using a graphing calculator or graphing software.
Lessons
Work through them in order.
About lesson progress
Lesson marks record what you have studied. Checks show understanding separately. Guest marks come from this device.
- Function rules and natural logarithmsConnect function rules, exponential/logarithm graphs and valid log algebra to growth, decay and fitted model parameters.
- Quadratic formula and modelsIdentify coefficients, retain both possible roots and reject values only for a stated domain reason.
- Completing the square and fractional quadraticsChoose an exact quadratic method and distinguish solving from rewriting a function.
- Quadratic extrema and signConnect completed squares and discriminants to strict sign conditions, model restrictions and feasible quadratic extrema.
- Discriminants, intersections and inequalitiesConnect substitution, root counts and tangency to sign intervals, number-line solutions and parameter cases that change degree.
- Graph features and numerical solutionsInterpret graph features, numerical boundaries and model equations, then check precision and feasible solutions.