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MAMS MATH
Function Toolbox
Ryan Smolenski, Group C
One of the goals for this course is for the student to be able to work with the relationship between variables. One of the most efficient ways to recognize
this relationship is by graphing. There are three term used when graphing a particular relationship that is (for the most part) continuous: draw, sketch, and
graph.
Draw is used when a quick picture of the relationship is all that is desired. No scales are used on the axes, but the axes are labeled.
Sketch is used for a more accurate representation. Scales and labels are provided on the axes, but no point-by-point plotting is used.
Graph is used for the most accurate representation. Scales and labels are provided on the axes, and point-by-point plotting is used.
The following toolbox of functions is provided for an overview of the functions we will be working with. It is by no means and exhaustive listing - feel
free to add to this list as the year progresses.
Algebraic Functions
Constant Function
Parent function: f[x] = n, where n Î R
Graph
Plot@f@xD=5,8x,-10,10<D
10
8
6
4
2
-10
-5
5
10
Domain
{x: x e R}
Range
5 in this example (whatever “n” is, where n e R)
Concavity
None
Increasing/Decreasing
Constant
Odd/Even
Even (reflects over y axis)
Math Modeling
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Mr. Barys
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Linear Function
Parent function: f[x] = x
Graph
Plot@f@xD=x,8x,-10,10<D
10
5
-10
-5
5
10
-5
-10
Domain
{x: x e R}
Range
{y: y e R}
Concavity
No
Increasing/Decreasing
Increasing
Odd/Even
Odd (reflects over the y and x axis)
Power Function
n
Parent function: f[x] = x , where n Î R
Graph(s)
Plot@f@xD=x^3, 8x,-5,5<D
100
50
-4
-2
2
4
-50
-100
Domain
{x: x e R}
Math Modeling
Mass Academy
Mr. Barys
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Range
{y: y e R}
Concavity
Yes. It is concave down for all negative x values, and concave up for all positive x values.
Increasing/Decreasing
Increasing
Odd/Even
Odd (reflects over the y and x axis)
Square Root Function
Parent function: f[x] =
x
Graph
PlotBf@xD=
x ,8x,-10,10<F
3.0
2.5
2.0
1.5
1.0
0.5
-10
-5
5
10
Domain
0 £ x € +¥
Range
0 £ y € +¥
Concavity
Yes. It is concave down for all positive x values.
Increasing/Decreasing
Increasing
Odd/Even
Neither (does not reflect)
Reciprocal Function
Parent function: f[x] =
1
x
Graph
Math Modeling
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Mr. Barys
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Plot@f@xD=1•x,8x,-10,10<D
1.0
0.5
-10
-5
5
10
-0.5
-1.0
Domain
{x: x e R}, where x ¹ 0
Range
{y: y e R}, where y ¹ 0
Concavity
Yes. It is concave down for all negative x values and concave up for all positive x values.
Increasing/Decreasing
Both (decreases to the right of the y-axis, increases on the left)
Odd/Even
Odd (reflects over the y and x axis)
Absolute Value Function
Parent function: f[x] = Abs[x]
Graph
Plot@f@xD=Abs@xD,8x,-10,10<D
10
8
6
4
2
-10
-5
5
10
Domain
{x: x e R}
Range
0 £ y € +¥
Concavity
No
Increasing/Decreasing
Decreases at first, but after x=0, it increases
Math Modeling
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Mr. Barys
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Odd/Even
Even (reflects once over the y-axis)
Transcendental Functions
Circular Functions
Cosine
Parent function: f[x] = Cos[x]
Graph
Plot@f@xD=Cos@xD,8x,-10,10<D
1.0
0.5
-10
-5
5
10
-0.5
-1.0
Domain
{x: x e R}
Range
-1 £ y £ +1
Concavity
Yes. For example, it is concave down for -1/2þ up to +1/2þ, but it is concave up for 1/2þ up to 3/2þ, and the pattern continually oscillates.
Increasing/Decreasing
Both
Odd/Even
Even (reflects over y-axis)
Sine
Parent function: f[x] = Sin[x]
Graph
Math Modeling
Mass Academy
Mr. Barys
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Plot@f@xD=Sin@xD,8x,-10,10<D
1.0
0.5
-10
-5
5
10
-0.5
-1.0
Domain
{x: x e R}
Range
-1 £ y £ +1
Concavity
Yes. For example, it is concave down from 0 up to þ, but it is concave up for þ up to 2þ, and the pattern continually oscillates. If it starts with an even þ, it
will be concave down, and the opposite happens if it starts with an odd coefficient of þ.
Increasing/Decreasing
Both
Odd/Even
Odd (reflects over both axes)
Tangent
Parent function: f[x] = Tan[x]
Graph
Plot@f@xD=Tan@xD,8x,-10,10<D
6
4
2
-10
-5
5
10
-2
-4
-6
Domain
All numbers
Range
-¥ £ y £ +¥
Concavity
Yes. At intervals of þ, the concavity increases to the y’s infinity. However, in intervals of _þ/2, where _ is an odd number, the tan becomes undefined, and
continually goes up. Immediately after that, the function becomes defined again, where it is concave decreasing up until þ, and the cycle repeats.
Increasing/Decreasing
Mr. Barys
to infinity, where the angle is _þ/2 radians, where _ is anMass
oddAcademy
coefficient. It will decrease close to negative infinity for a very small number
after _þ/2 radians, and will increase to zero when the angle is þ radians.
Math
Modelingup
It increases
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Increasing/Decreasing
It increases up to infinity, where the angle is _þ/2 radians, where _ is an odd coefficient. It will decrease close to negative infinity for a very small number
after _þ/2 radians, and will increase to zero when the angle is þ radians.
Odd/Even
Odd (reflects over both axes)
Exponential Functions
Parent function: f[x] = b x
Graph(s)
Plot@f@xD=3^x,8x,1,4<D
80
60
40
20
1.5
2.0
2.5
3.0
3.5
4.0
3.0
3.5
4.0
Domain
1 € x € +¥;
Range
0 € y € +¥
Concavity
Yes, it is concave up for all numbers that are at least one.
Increasing/Decreasing
Increasing
Odd/Even
Neither (does not reflect)
Plot@f@xD=1•3^x,8x,1,4<D
0.30
0.25
0.20
0.15
0.10
0.05
1.5
2.0
2.5
Domain
0 € x € 1;
Math Modeling
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Mr. Barys
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Range
0 € y € +¥
Concavity
Yes, it is concave up for all numbers that are between 0 and 1.
Increasing/Decreasing
Decreasing
Odd/Even
Neither (does not reflect)
Logarithmic Function
Parent function: f[x] = Logb @xD
Graph(s)
Plot@f@xD=Log10@xD,8x,-3,10<D
1.0
0.5
-2
2
4
6
8
10
-0.5
Domain
0 € x € +¥
Range
-¥ £ y £ +¥
Concavity
Yes, it is concave down for all positive x numbers.
Increasing/Decreasing
Increasing, but only because the number is more than one.
Odd/Even
Neither (does not reflect)
Math Modeling
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Mr. Barys