Prerequisites for Prerequisites for CalculusCalculus

Coordinate GeometryCoordinate Geometry

IncrementsIncrements SlopeSlope Parallel and Perpendicular LinesParallel and Perpendicular Lines Standard equations of LinesStandard equations of Lines Regression analysisRegression analysis

Contd.Contd. Increments in x = Increments in x = ΔΔx= xx= x22 – – xx11

Slope = m = rise/run = (ySlope = m = rise/run = (y22 – – yy11)/(x)/(x22

– – xx11)) Two lines are parallel if mTwo lines are parallel if m1 1 = m= m22

Two lines are perpendicular if Two lines are perpendicular if mm11mm22 = -1 = -1

Equation of a vertical line Equation of a vertical line through (a,b) is x = athrough (a,b) is x = a

Equation of a horizontal line Equation of a horizontal line through (a,b) is y = bthrough (a,b) is y = b

Contd.Contd. Point-slope equation y = m(x – Point-slope equation y = m(x –

xx11) + y) + y11

Slope-intercept equation y = mx Slope-intercept equation y = mx + b+ b

General equation of line Ax + By General equation of line Ax + By = C= C

If A(2, 3) and B(5, -7), find If A(2, 3) and B(5, -7), find Equation of line vertical through Equation of line vertical through

AA Equation of line through A and BEquation of line through A and B Equation of line perpendicular to Equation of line perpendicular to

AB passing trough B AB passing trough B

Functions and GraphsFunctions and Graphs

Function – Vertical line testFunction – Vertical line test Domain and RangeDomain and Range Recognizing graphs: Linear, Recognizing graphs: Linear,

Quadratic, Cubic, Rational, Quadratic, Cubic, Rational, Exponential, Logarithmic, and Exponential, Logarithmic, and Trigonometric Functions.Trigonometric Functions.

Even and Odd FunctionsEven and Odd Functions Piecewise functionsPiecewise functions Absolute value functionsAbsolute value functions Composite functionsComposite functions

Contd.Contd. A vertical line intersects the A vertical line intersects the

graph of a function in no more graph of a function in no more than one point.than one point.

All the x-values constitute the All the x-values constitute the DomainDomain

All the y-values constitute the All the y-values constitute the Range.Range.

An even function is symmetric An even function is symmetric about the Y-axis.about the Y-axis.

An odd function is symmetric An odd function is symmetric about the origin.about the origin.

Contd.Contd. In a piece-wise function, different In a piece-wise function, different

formulas are used to define the formulas are used to define the function in different parts of the function in different parts of the domain.domain.

Absolute value function y = Absolute value function y = abs(x) is defined as a piece-wise abs(x) is defined as a piece-wise function y = -x , x<0 and y = x, x function y = -x , x<0 and y = x, x ≥≥0 0

The composite function of g and The composite function of g and f is defined as f(g(x)) and f is defined as f(g(x)) and notation for this is notation for this is f f o o g.g.

Logarithmic and Exponential Logarithmic and Exponential Properties Properties log m + log n = log m + log n =

logmnlogmn log m – log n = log m – log n =

log(m/n)log(m/n) mlogx = logx^mmlogx = logx^m a^x a^x ·a^y = a^(x+y)·a^y = a^(x+y) a^x /a^x /a^y = a^(x– y)a^y = a^(x– y) (a^x)^y = a^xy(a^x)^y = a^xy a^-1 = 1/aa^-1 = 1/a

Functions and graphsFunctions and graphs

Linear : y = ax + b : Straight lineLinear : y = ax + b : Straight line Quadratic: y = ax^2 + bx + c Quadratic: y = ax^2 + bx + c

ParabolaParabola Absolute value: y = Absolute value: y = | x | | x |

: v – shaped: v – shaped Trigonometric FunctionsTrigonometric Functions Exponential and logarithmic Exponential and logarithmic

functionfunction

Answer the followingAnswer the following Factorize: Factorize:

Simplify:Simplify:

Graph the given Graph the given functions:functions:

Solve:Solve:

a a2 1 0 2 1 a a

a a

2

2

1 0 2 1

2 3 5

a y x

b y x

.

. s in

3 4

3

ax x

b x x

.

.

3 1

5

2 7

2

1 0 2 1 02

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