IB+Higher+Level+-+Functions+and+Equations

Return to Topic List

Although the Functions and Equations topic has been retained for the new syllabus, a bit of rearrangement has gone on, for clarity I will produce details of both. Under the old syllabi the chapters are arranged thus.

2.1

Concept of function //f: x |--> f(x)// : domain, range; image (value)

Composte functions //f o g// ; identity function

Inverse function //f// -1

2.2

The graph of a function; its equation //y = f(x)//

Function graphing skills:

use of a GDC to graph a variety of functions;

investigation of key features of graphs.

Solution of equations graphically.

2.3

Transformation of graphs: translations; stretches; reflections in the axes

The graph of //y = f -1 (x)// as the reflection in the line //y = x// of the graph of y = f(x)

The graph of //y = 1/f(x)// from //y = f(x)//

The graphs of the absolute value functions, //y = |f(x)|// and //y = f(|x|)//

2.4

The reciprocal function: //x |--> 1/x,// (x is not equal to 0): its graph; its self inverse nature

2.5

The quadratic function //x |--> ax// 2 //+ bx + c// : its graph, y-intercept //(0,c)//.

Axis of symmetry //x = - b/2a//

The form //x |--> a(x-h)// 2 //+ k// : vertex //(h,k)//.

The form //x |--> a(x-p)(x-q)// : x-intercepts //(p,0)// and //(q,0)//.

2.6

The solution of ax 2 + bx + c = 0, (a is not equal to 0).

The quadratic formula.

Use of the discriminant = b 2 - 4ac.

2.7

The function: x |--> a x, a > 0.

The inverse function x |--> log a x, x > 0.

Graphs of y = a x and y = log a x.

Solution of a x = b using logarithms.

2.8

The exponential function x |--> e x.

The logarithmic function x |-- In x, x > 0

2.9

Inequalities in one variable, using their graphical representation.

Solution of //g(x) >= f(x)// where //f,g// are linear or quadratic.

2.10

Polynomial functions.

The factor and remainder theorems, with application to the solution of polynomial equations and inequalities.

Under the new syllabus there are only seven chapters broadly covering the same material, but arranged thus.

2.1

Concept of function //f: x |--> f(x)//.

Domain, range; image (value)

Odd and even functions.

Composite functions //f o g//.

Identity function

One- to- one and many-to-one functions.

Inverse function //f// -1, including domain restriction. Self-inverse functions.

2.2

The graph of a function; its equation //y = f(x)//

Investigation of key features of graphs, such as maximum and minimum values, intercepts, horizontal and vertical asymptotes, symmetry and consideration of domain and range.

The graphs of the function //y = |f(x)|// and //y = f(|x|).//

The graph of //y = 1/f(x)// given the graph of //y = f(x)//

2.3

Transformation of graphs: translations; stretches; reflections in the axes

The graph of //y = f -1 (x)// as the reflection in the line //y = x.//

2.4

The rational function //x|--> (ax+b)/(cx+d)// and its graph.

The function //x|-->a x, a > 0 ,// and its graph

The function //x |--> log a x, x > 0 ,// and its graph

2.5

Polynomial functions and their graphs.

The factor and remainder theorems.

The fundamental theorem of algebra.

2.6

Solving quadratic equations using the quadratic formula.

Use of the discriminant //= b 2 - 4ac// to determine the nature of the roots.

Solving polynomial equations both graphically and algebraically.

Sum and product of the roots of polynomial equations.

Solution of //a x = b// using logarithms.

Use of the technology to solve a variety of equations, including those where there is no appropriate analytic approach.

2.7

Solutions of //g(x) >= f(x)//

Graphical or algebraic methods, for simple polynomials up to degree 3.

Use of technology for these and other functions.

Pages relating to Functions and equations
tangents to quadratics to be found. || 2.6 || J. Gregg ||
 * **Page Name** || **Description** || **Chapter ref.** || **Added by** ||
 * Finding Tangents || A GeoGebra file to generate questions which require