Lesson 2
Graphs and Estimation (Part I)
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Algebra and Calculus 2-1
Lesson 2 Overview
What is a graph?
Tables of values
Plotting the graph
Polynomials
Sketching curves
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Algebra and Calculus 2-2
What is a graph?
Graph is a visual method of showing
the behaviour of a function
On occasion, a graph may be the only
way to describe a function (a sensor
may monitor a patient's heartbeat - the
trace on a screen is a graph, but no
one could find an equation to give the
heartbeat at any particular instant)
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Algebra and Calculus 2-3
Tables of values
To plot a graph  Need a set of points
 Need a range of values of x and y
E.g. y = 3x+1
X
1
2
3
y
4
7
10
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Algebra and Calculus 2-4
Plotting the graph
From Tables of values, we can easily
draw points with co-ordinates (xi,yi) on
the axes.
y
yi
xi
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x
Algebra and Calculus 2-5
Polynomials
A function f(x) is a polynomial
expression if it can be written in the
form
f(x)=a+bx+cx2+dx3+ex4+....
where a, b, c, d, e etc are all constant
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Algebra and Calculus 2-6
Sketching curves
Linear curves
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Where it crosses the x-axis (when f(x) = 0)
Where it crosses the y-axis (when x = 0)
That it is a straight line
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Algebra and Calculus 2-7
Sketching curves
Quadratic curves




Where it crosses the y-axis (when x = 0)
The general shape, form the sign of the
coefficient of x2
Where it crosses the x-axis, by solving the
equation f(x) = 0
The maximum/minimum point when 2ax +
b = 0, that is at x = -b/2a and f(-b/2a)
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Algebra and Calculus 2-8
Sketching curves
Cubic curves


ax3 + bx2 + cx + d
Find where it crosses the y-axis
(when x = 0)
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Algebra and Calculus 2-9
Exponential curves
The curve must go through (0,1)
The whole graph is bound to be
above the x-axis
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Algebra and Calculus 2-10