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MAT091 :: Lecture Note :: Week 12
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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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Two lines that are not parallel can intersect. If the equations for the lines are known, then the point of intersection can be determined as follows.
line A: y = 2x + 2 line B: y = 5x + 5 set the lines equal to one another... 2x + 2 = 5x + 5 solve for x... 2x + 2 + 2 = 5x + 5 + 2 # add 2 to both sides 2x = 5x + 3 2x + 5x = 5x + 3 + 5x # add 5x to both sides 3x = 3 3x / 3 = 3 / 3 # divide both sides by 3 x = 1 substitute for x in one of the line equations and evaluate... y = 2(1) + 2 = 2 + 2 = 0 point of intersection: (1, 0) check answer: plug (1, 0) into the other line equation and see if it works... y = 5(1) + 5 = 5 + 5 = 0
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A "system" of equations is a collection of equations that are treated as an atomic unit. In other words, a system of equations is used when dealing with more than one variable.
A: 3x + 3y = 0 B: 12x + 2y = 20 pick one of the equations... pick A solve equation A for y 3x + 3y = 0 3x + 3x + 3y = 0 + 3x 3y = 3x 3y / 3 = 3x / 3 y = x substitute for y in the equation B... 12x + 2(x) = 20 solve equation B for x... 12x + 2x = 20 10x = 20 10x / 10 = 20 / 10 x = 2 pick one of the equations... pick A substitute for x in the equation A... 3(2) + 3y = 0 solve equation A for y... 3(2) + 3y = 0 6 + 3y = 0 6 + 6 + 3y = 0 + 6 3y = 6 3y / 3 = 6 / 3 y = 2 the solution is the point (2, 2)Another system of linear equations.
Two numbers have a sum of 55 and a difference of 9. Find the numbers. let x and y be the two unknown numbers form two equations... A: x + y = 55 B: x  y = 9 add the two equations together... x + y = 55 + x  y = 9 ============= 2x = 64 solve for x... 2x = 64 2x / 2 = 64 / 2 x = 32 pick an equation, substitute for x, solve for y pick A 32 + y = 55 32 + 32 + y = 55 + 32 y = 23 solution: (32, 23) [exercise] Redo this problem using substitution.Here is another system of linear equations that is solved using the addition method.
A: 5x + y = 10 B: x + y = 20 multiply both sides of B by 1... 1(x + y) = 20(1) x  y = 20 add the two equations... 5x + y = 10 + 1x  y = 20 =============== 4x = 10 solve for x... 4x / 4 = 10 / 4 x = 2.5 pick an equation and solve for y... pick B 2.5 + y = 20 y = 22.5 solution: (2.5, 22.5) subtitute the solution into A and see if it works... 5(2.5) + 22.5 = 10 12.5 + 22.5 = 10 it works!Is a system of equations necessary to solve the following problem?
Ten less than half of a number is thirty. What is the number?No, because there is only one variable.
n/2  10 = 30 n/2  10 + 10 = 30 + 10 n/2 = 40 n/2 * 2 = 40 * 2 n = 80Here is a exercise that can be solved using a system of equations.
A new nightclub had a grand opening in Scottsdale. First night admission was $4 for females and $6 for males. 90 people attended the grand opening and the nightclub collected $460. How many females and how many males attended the grand opening?let 'f' represent # of females and let 'm' represent # of males A: f + m = 90 B: $4(f) + $6(m) = $460 pick A and solve for 'f'... f = 90  m substitute 'f' into B... 4(90  m) + 6m = 460 distribute the 4, combine liketerms, solve for 'm'... 360  4m + 6m = 460 360 + 2m = 460 360 + 360 + 2m = 460  360 2m = 100 (1/2)2m = 100(1/2) m = 50 substitute m = 50 into A and solve for 'f'... f + 50 = 90 f = 40 There were 40 females and 50 males the grand opening.Education.Yahoo.com:: Solving Linear Systems
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The "intersection" feature of the TI83 calculator can be used to find the solution to a system of linear equations.
press Y= Y_{1} press CLEAR, if necessary enter equation: X is entered using key labeled X,T,theta,n 2X + 3 ENTER cursor at Y_{2}, press CLEAR if necessary enter equation: (1/3)X  4 ENTER press GRAPH press 2ND TRACE 5 (intersect) First curve? ENTER Second curve? ENTER Guess? ENTER Intersection x = 3 and y = 3 (3, 3)Substitute
x = 3
into both equations and test if(3, 3)
is a solution.f(x) = 2x + 3 f(3) = 2(3) + 3 = 6 + 3 = 3 check f(x) = (1/3)x  4 f(3) = (1/3)(3)  4 = 1  4 = 3 check
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