IB+Standard+Level+-+Functions+and+equations

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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)

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 |--> ax, a > 0.

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

Graphs of y = ax and y = logax.

Solution of ax = b using logarithms.

2.8

The exponential function x |--> e x.

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

==

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

2.1

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

Domain, range; image (value)

Composite functions.

Identity function

Inverse function //f// -1

2.2

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

Function graphing skills:

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

Use of technology to graph a variety of functions, including ones not specifically mentioned.

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

2.3

Transformation of graphs:

Translations: //y = f(x) + b ; y = f(x-a)//

Reflections (in both axes): //y = -f(x) and f(-x).//

Vertical stretch with scale factor //p: y = pf(x).//

Stretch in the //x-//direction with scale factor 1///q: y = f(qx)//

Composite transformations.

2.4

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-p)(x-q)// : x-intercepts //(p,0)// and //(q,0)//.

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

2.5

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

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

Vertical and horizontal asymptotes

2.6

Exponential functions and their graphs: //x|-->a x, a > 0 , x |--> e x .//

Logarithmic functions and their graphs //x |--> log a x, x > 0 , x|--> In x , x > 0 .//

Relationships between these functions: //a x =e xIna ; log a a x = x ; a log // a // x = x, x > 0 .//

2.7

Solving equations, both graphically and analytically.

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

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

The quadratic formula

The discriminant = b 2 - 4ac and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.

Solving exponential equations.

2.8

Applications of graphing skills and solving equations that relate to real-life situations.

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