R00501 Functions Overview
R00501 Functions Overview
Overview
This is ground Zero. If you don't understand the topics on this page, you have to stay here until you do. An understanding of functions is a grounding for all algebra and programming after this. It is bummer of you don't know these answers. In this section you are basically tested on algebra from grade 10 and grade 11.
If you don't understand the topics here start with Revision of Grade11
Students must get full marks for the first 10 questions of Trial Exam. The student below cannot advance past this lecture. Without this grounding, your ability to grasp further concepts will be severely restricted.
Goals
IEB/NSC will test you how you can solve for a variable and draw graphs of given functions.
You need to know Linear Functions, Parabola, Hyperbole and Exponential.
| Linear Functions | y = mx + c |
| Parabola | y = a( x+p)2 + q |
| Hyperbole | y = a/(x+p) + q |
| Exponential | y = a . b x+p + q |
Identify each type of equation
Draw a rough sketch of each Graph
Calculate possible x and y-intercepts,
asymtotes, maxima, minima, inflexion points
Quadratics
The Parabola
Three forms of a parabola
| y = a (x-p)2 + q | vertex form |
| y = a x2 + bx + c | standard form |
| y = a (x- r1) (x-r2) | intercept form |
a Determines the shape
if a > 0 Parabola is concave UP - smile
if a < 0 Parabola is concave DOWN – grim
The further a from 0 the thinner the graph
Vertical Shift
The greater the value of q or c the greater
the vertical shift.
Positive values of q or c
move the graph UP,
Negative values of q or c
move the graph DOWN
Axis of symmetry
A Parabola is always vertically symmetrical.
The axis of symmetry is half way
between the Roots (where y = 0)
The Turning Point
The turning point is on
the Axis of Symmetry.
In vertex form this point is (p,q)
Hyperbola
The Hyperbola
The Hyperbola
\(y = {\color{red}{ a} \over x - \color{green}{ p}} + \color{blue}{ q} \)
a determines shape
q determines vertical shift
p determines horizontal shift
Shape
a > 0
Hyperbole lies in 1st and 3rd quadrant
a < 0
Hyperbole lies in 2nd and 4th quadrant
Axis of Symmetry
Hyperbole is Symmetrical about :
y = x - p + q and y = x + p + q
Domain and Range
Domain
x ε ℝ or x ε (- ∞ ; ∞) and x ≠ p
Range
y ε ℝ or y ε (- ∞ ; ∞) and y ≠ q
Intersection with Axis of Symmetry
a > 0
Intersects y = x - p + q at
(√a + p ; √a + q)
and
(-√a + p ; -√a + q)
a < 0
Intersects y = x - p + q at
(-√a + p ; √a + q)
and
(-√a + p ; -√a + q)
External Links
| Siyavula | |
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| BBC | Functions - Composite Functions Functions - Advanced Composite Functions Functions - Graphs of Equations |
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| Brian McLogan The Maths Page |
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| Mathplanet | Solve Equations and Simplify expressions |
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Mindshare
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Solving a Quadratic Equation with Complex Roots
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| Exam Solutions..... | Identity or equation | |
| Varsity Tutors | Equations of a Parabola in Vertex Form | |
| Meta Calculator | App for plotting graphs | |
Write down the equations of any asymptotes
KahnWrite down the equation of the axis of symmmetry of a parabola and the coordinates of the turning point.
Exercises
Exercises
Here is a good way to start using you knowledge of linear functions to plan your own progress to an A pass.
We assume that your progress is a linear function.You will need to progress from your current level to 90% in the number of study weeks remaining. You must be able to do this over Both papers. Take you current score for Trial Exam and get the formula for a straight line form current level to 90% in the number of weeks left.
For example is there are 6 weeks left and your current score is 60, you will need to improve by 5% per week to get to 90% by exam time. Follow the exercises in Marking Criteria
| That Quizz | Difference of Two Squares | |
| That Quizz |
Completing the Square | |
| That Quizz | and solving by factoring | |
| That Quizz |
Exercises
Do these revision exercises on paper BEFORE looking at the answers
1. Sketch the Graph of y = 1/2 x + 3
Cowan Academy
2. Sketch the Graph of y = -1/2 x
Cowan Academy
3. Sketch the Graph of y = 1/4 x - 2
More Overview
Students - you were not paying attention last year, fix it NOW
Wolves - You have no place being a wolf if you are struggling with this. You need to sign up for an instructor to help you. You have definitely misrepresented your abilities to us. If you go past this point without scoring more than 90% in this sections in the Exams, we will hunt you down and shoot you.
Adults - You had a mispent youth or you have indulged too much in wine women and song. Take your time over this lecture.
Parent - You have to cover this now. If you move on without getting it, your child is likely to run away from you through the tree. Stick with them
Lions - If you have never encountered pre Algebra, spend a time on this lecture (Schools take a year to teach it). Really focus on the Revision of Grade11 and Exercises. It will not help you to advance before you have waxed this lecture. You may have to slow down on this lecture to catch up faster later. Getting it right will make everything that comes next much easier
Instructors, Teachers, Manager - Functions is where the rubber hits the road. Do not advance Students past this until they have it. You have to be cruel to be kind. If they are struggling here, the read lights should start flashing. They are unlikely to get to 90% in less than 12 weeks. If they stick here, this may be the moment to collect your axe and take them out for a walk up the mountain. The gods are hungry for a sacrifice.