Function Family Album
73
Quadratic Function
Cube Root Function in Real Life
Cube Root Function
Quadratic Function in Real Life
The photo on the right resembles a cube root function. It has a point of symmetry and appears to extend on into infinity and negative infinity.
The photo on the right resembles a quadratic function. It has a general U-shape and appears to have an axis of symmetry and a vertex.
Max & Min Values: Infinity and negative infinity
Vertex: None
Asymptotes: None
X Intercept: Set y to 0
Y Intercept: Set x to 0
Domain & Range: All Real Numbers
Symmetry: Point Symmetry, estimate point and test
Parent Function: y= cube root of x
Standard Form: y=a(x-c)^1/3 +d
Transformations:
Max Value: -b/2a, a<0
Min Value: -b/2a, a>0
Vertex: (h,k)
Asymptotes: none
X Intercept: Use (-b +/- sq. root of b^2-4ac)/2a
Y Intercept: Set x to 0 and solve
Domain: All real numbers
Range: All real numbers greater/equal to the y-intercept.
Symmetry: Axis of Symmetry, solve -b/2a
Parent Function: y=x^2
Standard Form: y= Ax^2 +Bx+C or y=a(x-h)^2 +k
Transformations:
Square Root Function
y=cube root of negative x, flips it across the y axis
y= cube root of x plus 2, moves function up 2 units
y= cube root of x minus 2, moves function down 2
y= cube root of (x+2), moves function left 2 units
y= cube root of (x-2), moves function right units
y= 3*cube root of x, stretches the graph
y= 1/3* cube root of x, shrinks the graph
y=x^2 +2, raises it up 2 units
y=x^2 -2, lowers it by 2 units
y=(x+2)^2, moves it left 2 units
y=(x-2)^2, moves it right 2 units
y=-x^2, flips function over x axis
Max & Min Values: No relative max/min
Vertex: (h,k)
Asymptotes: None
X Intercept: x>/= h
Y Intercept: y>/= k
Domain & Range: x>/=h, y>/=k
Symmetry: No symmetry
Parent Function: y= sq. root of x
Standard Form: y= (sq. root of x-h)+k
Transformations:
Greatest Integer Function
Max & Min Values: None
Vertex: None, but there is a starting point at (h,k)
Asymptotes: None
X Intercept: (n, n+1)
Y Intercept: k
Domain & Range: All Real Numbers, all integers
Symmetry: None
Parent Function: y= [[x]]
Standard Form: y=a[b(x-h)]+k
Transformations:
y= - sq. root of x, reflects across the x axis
y= sq. root of -x, reflects across y axis
y= sq. root of x-2, moves 2 units to right
y= sq. root of x+2, moves 2 units to left
y= (sq. root of x)+2, moves 2 units up
y= (sq. root of x)-2, moves 2 units down
a>1, stretches. a<1, flattens
Linear Functions
Square Root Function In Real Life
The photo on the right appears to model a square root function. It has a vertex/starting point and has the general shape of a square root function. It does not appear to have any asymptotes.
y=-[x], reflects across the x axis
y=[-x], reflects across the y axis
y=2[x], increases distance between steps
y=.5[x], decreases distance between steps
y=[x+2], moves graph left 2 units
y=[x-2], moves graph right 2 units
y=[x]+2, moves graph up 2 units
y=[x]-2, moves graph down 2 units
Changing b changes the length of each step
Exponential Function in Real Life
The photo on the right resembles a exponential function by the way its height increase dramatically and the way the line on the left part still decreases, giving the line an L like shape.
Max & Min Values: None, as they stretch on forever
Vertex: None
Asymptotes: None
X Intercept: Set y=0, solve for x
Y Intercept: Set x=0, solve for y
Domain & Range: All Real Numbers
Symmetry: None
Parent Function: y=x
Standard Form: Ax+By=C or y=mx+b
Transformations:
Linear Absolute Value Function
Exponential Function
Greatest Integer Function in Real Life
Linear Absolute Value in Real Life
Max & Min: When the absolute value in the function equals 0, then k
Vertex: (h,k)
Asymptotes: None
X Intercept: Solve as if y equals 0
Y Intercept: Solve as if x equals 0
Domain & Range: All real numbers, for y, greater than or equal to k
Constrained Domain: (example)1</= x </=2
Symmetry: x=h
Parent Function: y= |x|
Standard Form: y=|x-h|+k
Transformations:
The photo on the right resembles a step/greatest integer function. Each one of the steps are of equal length and equal distance apart.
Max & Min: None
Vertex: None
Asymptotes: y=b^x+1, y=1. If the function doesn't have addition, the asymptote is 0
X Intercept: None
Y Intercept: Substitute 0 for x
Domain: All real numbers
Range: y>the asymptote
Symmetry: None
Parent Function: y=b^x, where |b|>0
Standard Form: y= a*b^x
Transformations:
The photo on the right appears to resemble a linear absolute value function. It has a axis of symmetry and a vertex. It also has the V-shape that all linear absolute value functions have.
y=x+1 (Moves function up 1)
y-x-1 (Moves down 1)
y=2x (Becomes steeper)
Y=.5x (Becomes flatter)
Y=-x (Flips and slopes downward)
b=2
y= -1|x|, flips the function on the x axis
y= 2|x| narrows the function
y= |x-1| moves function right 1 unit
y= |x+1| moves function left 1 unit
y=|x| +1 moves function up 1
y=|x|-1 moves function down 1
0<b<1, flips function across y axis
y=b^x +1, raises it up 1 unit
y=b^x -1 lowers it 1 unit
y= 2*b^x steepens the graph
y= .5*b^x flattens the graph
y= -3*b^x flips function over x axis
Cubic Function
Linear Function in Real Life
Max & Min Values: None
Vertex: None
Asymptotes: None
X Intercept: Set y to 0
Y Intercept: Set x to 0
Domain & Range: All Real Numbers
Symmetry: Point Symmetry, estimate point and plug it into the factored out version of the function
Parent Function: y=x^3
Standard Form: y=ax^3+bx^2+cx+d
Tangent Line: (y-y0)=f'(x-value)(x-x0)
Transformations:
The bottle caps on the right reflect a linear function because it is a straight line and the heights(y-values) increase with length (x-values).
Cubic Function in Real Life
y=x^3 + 2, moves graph up 2 units
y=x^3 -2, moves graph down 2 units
y= (x+2)^3, moves graph left 2 units
y=(x-2)^3, moves graph right 2 units
A small negative number for 'a' closes the graph
A large positive number for 'b' opens the graph
-f(x) reflects it across the x-axis
f(-x) reflects it across the y-axis
Works Cited
The photo on the right resembles a cubic function. It has a point of symmetry and has the shape of a cubic function. It passes the vertical line test.
- http://www.mathops.com/free/a1av002.php?358
- http://www.freemathhelp.com/forum/threads/49933-Finding-X-and-Y-Intercepts-w-Absolute-Value-y-x-5
- http://www.ck12.org/book/CK-12-Math-Analysis/r27/section/1.5/
- http://tutorial.math.lamar.edu/Classes/CalcI/MinMaxValues.aspx
- http://americanomath.wikispaces.com/Cubic+Functions
- https://www.khanacademy.org/math/algebra2/functions_and_graphs/piecewise-functions-tutorial/v/graphs-of-absolute-value-functions
- http://www.educadores.diaadia.pr.gov.br/arquivos/File/2010/artigos_teses/EDUCACAO_E_TECNOLOGIA/CUBICS.PDF
- http://www.analyzemath.com/Graphing/graphing_cubic_function.html
- http://www.matematicasvisuales.com/english/html/analysis/derivative/cubic.html
- http://www.algebra.com/algebra/homework/quadratic/Quadratic_Equations.faq.question.11060.html
- https://www.physicsforums.com/threads/finding-a-tangent-in-a-cubic-function.408091/
- http://fcw.needham.k12.ma.us/~jessica_kondrat/fov1-00100c91/fov1-00100c96/5%3A4%20per6.pdf
- http://msenux.redwoods.edu/IntAlgText/chapter9/section1.pdf
Works Cited
- http://www.analyzemath.com/function/cube_root_function.html
- http://www.onlinemathlearning.com/quadratic-square-root-cube-root-functions.html
- http://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGTypeSquareRoot.html
- http://www.math.fsu.edu/~everage/MAC1105/LibFunc.pdf
- http://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGTypePiecewise.html
- http://www.docstoc.com/docs/8859415/Greatest-Integer-Function
- http://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGTypeSquareRoot.html
- http://www.shmoop.com/linear-equations/absolute-value-functions.html
- https://www.desmos.com/calculator
- http://altbeta.icoachmath.com/math_dictionary/greatest_integer_function.html
- http://math.about.com/od/algebra1help/a/Func.htm
- http://mathbitsnotebook.com/Algebra1/FunctionGraphs/FNGTypePiecewise.html