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MAT091 :: Lecture Note :: Week 06
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When it comes to graphing functions, tables and other forms of data, input values are scaled along the horizontalaxis and output values are scaled along the verticalaxis.
When it comes to naming variables, the letter
x
is often used to represent the input variable and the lettery
is used to represent the output variable; however, any letters can be used when it comes to naming input and output variables.Table A =============================================================== input: year (y)  2001  2002  2003  2004  2005  2007  output: # xrays (x)  55 43 81 24 43 61 The # of xrays is a function of the year; i.e. year is the input and # of xrays is the output. For this function, the input variable is named 'y' and the output variable is named 'x'. Table A expressed using function notation. f(y) = x f(2001) = 55 f(2002) = 43 f(2003) = 81 ... f(2000) = undefined [cannot extrapolate] f(2006) = undefined [cannot interpolate] Table B ============================================================== input: hours (h)  1  2  3  4  5  6  7  output: minutes (m)  60  120  180  240  300  360  420 Number of minutes is a function of number of hours. One hour is 60 minutes, two hours is 120 minutes, etc. Table B expressed using function notation. g(h) = m = 60(h) g(1) = 60 g(4) = 240 g(7) = 420 g(2.5) = 150 [interpolation] g(5.25) = 315 [interpolation] g(0) = 0 [extrapolation] g(8) = 480 [extrapolation]
[definition]
A function is a "relation for which each element of the domain corresponds to exactly one element of the range."
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Functions are sometimes categorized as either increasing or decreasing.
An increasing function is a function whose output values increase (i.e. get larger) when the inputs increase.
f(0) = 5 f(1) = 10 f(2) = 15 f(3) = 20 f(x) appears to be an increasing functionA decreasing function is a function whose output values decrease (i.e. get smaller) when the inputs increase.
g(0) = 100 g(1) = 90 g(2) = 80 g(3) = 70 g(x) appears to be a decreasing functionA constant function
f(x) = k
(where 'k' is a constant) is neither increasing nor decreasing.The identity function
f(x) = x
is an increasing function.Some functions can be both increasing and decreasing. For example, over an interval A, a function might be increasing, while it decreases over an interval B. Review the absolute value function.
abs(x) is decreasing from (INF, 0) abs(x) is zero at 0 abs(x) is increasing from (0, INF)Technical definitions for increasing and decreasing functions.
increasing function: ==================== Function f(x) increases on an interval I if f(b) > f(a) for all b > a, where a,b are in I. decreasing function: ==================== Function f(x) decreases on an interval I if f(b) < f(a) for all b > a, where a,b are in I.
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MW.com defines linear as follows.
"of, relating to, resembling, or having a graph that is a line and especially a straight line" [...and...] "having or being a response or output that is directly proportional to the input"Observe that the word linear contains the word line.
A linear function is a function is a "first degree polynomial function of one variable. These functions are called 'linear' because they are precisely the functions whose graph in the Cartesian coordinate plane is a straight line."
The following are equations for a line.
slopeintercept form: y = mx + b ... used when slope and yintercept are known pointslope form: y  y_{1} = m(x  x_{1}) ... where (x_{1}, y_{1}) is a point on the line having slope m standard form: Ax + By = CSlope is another way of saying "rate of change" and linear functions have a constant rate of change.
Slope is often described as the ratio "rise over run."
Linear functions with a positive constant rate of change are increasing functions. Linear functions with a negative constant rate of change are decreasing functions.
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The slope of a line is a measure of its average rate of change (steepness?). The slope also indicates if the line is increasing (uphill?) or decreasing (downhill?). In linear equations, slope is represented by the letter m.
The following is how to find the slope between two unique points on a line.
point 1: (x_{1}, y_{1}) point 2: (x_{2}, y_{2}) slope = m = (y_{2}  y_{1}) / (x_{2}  x_{1}) note: x_{1} ≠ x_{2}Slope represents the ratio of changeinoutput over changeininput.
m = Δy / Δx (change in output / change in input) Δ is the "delta" character Δy is read "change of" (Δy is read "change of" y) Δy = y_{2}  y_{1}Slope is often described as the ratio "rise over run."
rise m =  runSome slope notes.
horizonal lines have slope 0 (y = f(x) = k, where k is a consant) vertical lines have undefined slope (x = k, where k is a consant) parallel lines have equal slopes (m_{1} = m_{2}) perpendicular lines have negative inverse slopes (m_{2} = 1/m_{1})The following is a verbose form the slopeintercept equation.
output = slope * input + initial_value_if_any ...or... output = average_rate_of_change * input + initial_value_if_any
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We frequently need to know the points where lines (graphs in general) intersect (cross) the horizonalaxis and verticalaxis (often called the xintercept and yintercept, respectively).
The verticalintercept for a function is found by evaluating the function when its input is zero. The orderdpair for a verticalintercept will be
(0, something)
.The horizontalintercept for a function is found by setting its output to zero and finding what input value results in that output. The orderdpair for a horizonalintercept will be
(something, 0)
.f(x) = 5(x) + 3 verticalintercept (yintercept)... set the input to 0 and evaluate: f(0) = 5(0) + 3 = 3 verticalintercept is (0, 3) horizontalintercept (xintercept)... set the output to 0 and solve: 0 = 5(x) + 3 3 = 5(x) 3/5 = x horizontalintercept is (3/5, 0)Generalizations.
given f(x) = m(x) + b verticalintercept: (0, b) [notice input is 0] horizontalintercept: (b/m, 0) [notice output is 0]
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Memorize the following.
slope formula: m = (y_{2}  y_{1}) / (x_{2}  x_{1}) slopeintercept form: y = mx + b pointslope form: y  y_{1} = m(x  x_{1})
[exercise]
Find the equation of the line having a slope of5
that passes through the point(2, 8)
.step 1: substitute the slope 5 into y = mx + b y = 5(x) + b step 2: substitute 8 for y and 2 for x and solve for b 8 = 5(2) + b 8 = 10 + b 10 + 8 = 10 + 10 + b 2 = b step 3: rewrite the equation from step #1 and substitute 2 for b y = 5(x) + 2
[exercise]
Given the two points(1, 5)
and(3, 15)
, find the equation of the line.step 1: calculate the slope m (15  5) / (3  1) = 10 / 2 = 5 step 2: substitute one of the points and the slope into the pointslope form equation for a line y  5 = 5(x  1) # point (1, 5) selected; m=5 y_{1}=5, x_{1}=1 step 3: solve for y y  5 = 5x  5 # distributive property on rightside y = 5x  5 + 5 # add 5 to both sides y = 5x + 0 # this is slopeintercept formSince the 'b' value is zero we know that this line passes through point
(0, 0)
. Recall, the 'b' value is the yintercept (verticalintercept) and it is obtained by setting the input to zero.Solving this problem using the TI83 calcuator.
press ON press STAT press 1:EDIT if there are values in L1, then left arrow to L1 column, if necessary up arrow to L1 cell, if necessary press CLEAR press ENTER if there are values in L2, then up arrow to L1 cell right arrow to L2 cell press CLEAR press ENTER if necessary, use arrow keys to get to line 1 of L1 1 ENTER 3 ENTER right arrow 5 ENTER 15 ENTER There are now two point entered. L1  L2  1  5 3  15 press STAT right arrow to CALC press 4:LinReg(ax + b) home screen says: LinReg(ax + b) [cursor] press 2ND 1 (L1) press COMMA (,) press 2ND 2 press COMMA (,) press VARS right arrow to YVARS press 1:Function press 1:Y1 home screen says: LinReg(ax + b) L1,L2,Y1 [cursor] press ENTER home screen says: LinReg y = ax+b a = 5 b = 0 r^2 = 1 r = 1 press 2ND GRAPH to see a table press GRAPH to see a graph of the line press Y= to see the linear equation
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