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1 .3 Linear Equations in two Variables

Slope of a line is a ratio of the difference in the vertical distanceto the horizontal distance between two points on that line.Slope can be represented many ways, such as: 2 1

2 1

y yrise ym

run x x x

There are four possible slopes of lines:Positive, which increases from left to rightNegative, which decreases from left to rightZero, which is horizontal (constant function)

Undefined or no slope, which is vertical

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Ex. Find the slope of the line which passes through thepoints (7, -1 ) and (-1 ,1 )

8 2

Ex. (-5, 6) and (1 0,4)

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Parallel lines have the same slopeSlopes of perpendicular lines are opposites and reciprocals

of each other

Ex. Find the slope of a line to the line which passes

through (2,-1 ) and (5,8).

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Three forms of equations of lines:Standard form Ax + By = CSlope-intercept form y = mx +bPoint-slope [you can use any point on the line for this]

To write the equation of a line, the minimum informationneeded is the slope and at least one point

1 1( )y y m x x

Ex. Write the equation of the line which passes through (4,-2)and is parallel to 3x - 4y = 1 0.

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Regression - fits data to an equation

1 . Enter the data into a blank tableSTAT Edit

2. Graph the scatterplot of the data

y = Turn PLOTon ZOOM 9 GRAPH

3. Fit to an equation: STAT CALC LinREG L1 , L2, Y1 ENTER

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1 .4 FunctionsA function of y in terms of x is a relation where every x value is paired

with exactly (one and only one) y value.

oNumerically as in a table or a list of points any repeated x values must bepaired with the same y values

oGraphically, a function must pass the vertical line test. As an imaginary

vertical line moves across the coordinate plane horizontally, the

graph of a function cannot intersect the vertical line more than

once.

oAlgebraically, an equation of a function must be able to be solved for ysuch that each x value inputted into the equation results in one

output.

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Ex. Is x + y = 1 6 a function?22

Function notation - solve for y, then replace y with f(x)

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Piecewise functions are literally pieces of functions put together to

make one new function. Here is an example of a piecewise

function to show how to write the algebraic representation of it:

and here is the graphical representation of it:

2

9, 3

( ) , 3 2

13, 2

2

x

f x x x

x x

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Domain of a function is the set of all x values (input) that havecorresponding real y values

Domain restrictions:

Square roots ( or any root with even index) Rational expressions Logarithmic functions (Ch. 3) Some trig functions (Ch. 4-6)

Ex. Find the domain of:

1 . f(x) =

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