About this topic
Before you begin
Connect exponential and logarithmic models with graph features, numerical solutions and exact quadratic methods.
Learning goals
- Produce exact logarithmic values with valid real bases and arguments
- Produce complete real exponential equation results
- Produce calibrated exponential model parameters, predictions and target times
- Model and interpret exponential and logarithmic relationships
- Decide which relations give exactly one output for every input
- Combine natural logarithms into one logarithm on a stated domain
- Interpret a function as one output for each input
- Use logarithm laws with domain conditions
- Formulate, use and interpret a mathematical model
- Solve quadratic equations by suitable methods
- Derive complete parameter sets for a quadratic to be always positive or negative
- Interpret and draw graphs of functions
- Use quadratic structure and discriminants to reason about roots
- Derive complete parameter sets for the real roots of a quadratic
- Find the complete real solution set of a rational quadratic inequality
- Find every real solution of a rational line–quadratic system
- Solve simultaneous equations by justified methods
- Solve inequalities and represent their solution sets
- Solve a system of linear equations completely, including systems with no or infinitely many solutions
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.
Choose a lesson
Work through them in order.
- Not startedFunction rules and natural logarithmsNextConnect function rules, exponential/logarithm graphs and valid log algebra to growth, decay and fitted model parameters.
- Not startedQuadratic formula and modelsNextIdentify coefficients, retain both possible roots and reject values only for a stated domain reason.
- Not startedCompleting the square and fractional quadraticsNextChoose an exact quadratic method and distinguish solving from rewriting a function.
- Not startedQuadratic extrema and signNextConnect completed squares and discriminants to strict sign conditions, model restrictions and feasible quadratic extrema.
- Not startedDiscriminants, intersections and inequalitiesNextConnect substitution, root counts and tangency to sign intervals, number-line solutions and parameter cases that change degree.
- Not startedGraph features and numerical solutionsNextInterpret graph features, numerical boundaries and model equations, then check precision and feasible solutions.
- Not startedLinear systems from contextNextTurn three conditions into a system of linear equations, solve it with a graphing calculator and check for repeated or contradictory conditions.