FUNCTIONSp. 1

8.2 Represent basic functions (linear, quadratic, exponential, and reciprocal) and piecewise-defined functions (varying over subintervals of the domain) using and translating among words, tables, graphs, and symbols.

8.2 Represent basic functions listed above and the following advanced functions (cubic, quartic, logarithmic, square root, cube root, absolute value, and rational functions of the type usingand translating among words, tables, graphs, and symbols.

High School / Community College / EWU

a.Evaluate functions to generate a graph.

Core 1
Core 2
Make graphs and tables. Describe the way y changes as x increases.
(a)
(b)
Core 3
Integrated 3
Graph: Show vertical asymptotes and horizontal asymptotes, x-intercepts, and y-intercepts, and state the domain and range.
Graph: State the x-intercepts and y-intercepts, domain, range, maximum, and minimum.
Graphand identify maximum/minimum, y-intercept, and x-intercept
Graph and identify the x and y-intercepts
Algebra 2, CPM
Create a table to graph the following relations.
(a)
(b)
Algebra 2
It is not in our curriculum
Algebra 2/Trigonometry
Precalculus
Graph the function: / SFCC – Beginning & Intermediate Algebra
Graph the functionby using a table.
Graph the function by identifying the vertex, the intercepts and several other points on the graph.
SCC – Intermediate Algebra
Graph: / Basic Algebra
Graph the function by setting up tables of values. Explain why the graphs are different.
for
for any real number
Intermediate Algebra
Graph the function by identifying the vertex, the intercepts and several other points on the graph.
Precalculus I
Graph the following functions.
(a)
(b)
(c)

b.Describe relationships between the algebraic features of a function and the features of its graph and/or its tabular representation.

Core 1
Core 2
Look back over your discoveries in the experiments with various power models.
  • What patterns do you expect to find in tables and graphs of relations with equations of the formwhen a is some positive number? When a is some negative number?
  • What patterns do you expect to find in tables and graphs of relations with equations of the form when a is some positive number? When a is some negative number?
  • What patterns to do you expect to find in tables and graphs of power models when n is a positive even integer? When n is a positive odd integer?
  • How are the patterns in tables and graphs of power models different from those of linear and exponential models?
  • Be prepared to share you group’s conclusions with the entire class.
Core 3
Integrated 3
Graph: State: vertex, x and y intercepts, maximum and minimum. What effect does the 3 and 4 have on ?
See SFCC beginning and Intermediate Algebra and EWU Basic Algebra; We have similar questions on tests.
Algebra 2, CPM
What important points do you need to graph a parabola? Given , find all intercepts and the vertex. Then graph the parabola on a complete graph.
Algebra 2
It is not in our curriculum
Algebra 2/Trigonometry
Precalculus
Given that a>0, b>0, c>0, and ca, sketch a graph of the function . Be sure to include and label all intercepts and indicate the end behavior of the graph. Briefly explain your response / SFCC – Beginning & Intermediate Algebra
What does the value of a intell you about the graph of the function compared to the graph of
Match the following:




iVertex (2,-4) opens down
iiVertex (2,-4) opens up
iiiVertex (4,-2) opens down
ivVertex (4,-2) opens up
SCC – Intermediate Algebra
Find the vertex of by writing the function is the form / Basic Algebra
Find a, b, and c in the quadratic function . How does the value of a affect the graph? How does the value of b affect the graph? How does the value of c affect the graph?
Intermediate Algebra
Determine the absolute maximum/minimum of the function given in part (a).
Algebra Concepts
Find the symbolic form of a rational function whose graph has vertical asymptotes at and
Precalculus I
Let . How does the graph of change as
i) gets larger
ii) goes to negative infinity
iii) goes to infinity
iv) goes to negative infinity.

c.Use simple transformations (horizontal and vertical shifts, reflections about axes, shrinks and stretches) to create the graphs of new functions using linear, quadratic, and/or absolute value functions, cubic quartic, exponential, logarithmic, square root, cube root, absolute value, piecewise and rational functions of the type .

Core 1
Core 2
Core 3
Integrated 3
Using as the “parent” function, tell what happens to . Use words such as stretch, shrink, flip, slide, vertical translation, etc.
Using as the parent function, describe the transformations that occur to create the graph
Algebra 2, CPM
Algebra 2
We do this with linear and quadratic equations only
Algebra 2/Trigonometry
Precalculus
Describe the transformation of the function (where a>0) illustrated by each of the following:
(a)
(b)
(c)
(d)
(e) / SFCC – Beginning & Intermediate Algebra
Graph the following functions:
(a)
(b)
(c)
(d)
SCC – Intermediate Algebra
It is not in our curriculum / Algebra Concepts
Use the graph of and transformations to graph
Precalculus I
Explain how to graph the following functions by comparing to :
(a)
(b)
(c)
Use transformations to graph the following functions ,
Explain how to graph the following functions by comparing to :
(a)
(b)
(c)
(d)
(e)
Let have the graph shown below.

Match one of the following functions to each of the indicated graphs.
A.
B.
C.
D.
E.


d.Algebraically construct new functions using addition and subtraction (e.g., profit function), multiplication, division, and composition

Core 1
Core 2
Core 3
Integrated 3
Given and find the following: State the domain and range of the new function.
It is in our curriculum, but I skip it
Algebra 2, CPM
Algebra 2
It is not in our curriculum
Algebra 2/Trigonometry
Precalculus
Given and , find each of the following:
(a)
(b)
(c)
(d)
(e)
(f) / SFCC – Beginning & Intermediate Algebra
When a thermal layer is over a city pollutants are trapped and must disperse horizontally. Assume that a smokestack begins emitting a pollutant at 8 a.m. and that the pollutant disperses horizontally over a circular area. Let t represent time, in hours, since the factory began emitting pollutants and assume that the radius of the circle is 2t miles. Let prepresent the area of a circle of radius r. Find and interpret
SCC – Intermediate Algebra
It is not in our curriculum / Basic Algebra
Martha has a house-sitting business. She charges $50 to establish an account, then $20 per day to stay in client’s homes. Martha spends $8 per day in advertising and transportation costs. Write a cost function, , to determine Martha’s costs for x number of days. Write a revenue function, , to determine Martha’s revenue for x number of days. Write a profit function, , to determine Martha’s profit for x number of days.
Intermediate Algebra Example with profit
Precalculus I
If Revenue is given by and cost is given by find a function that represents the profit that the company makes when x units of product are made.
Let and
What is , and
If is the equation for the area of a rectangle with one side having length , what is length of the other side?

e.Given an algebraic representation of a rational function, find the intercepts, asymptotes (horizontal, vertical, and slant), and holes (discontinuities), then sketch the graph.

Core 1
Core 2
Core 3
Integrated 3
Graph and list all features (horizontal and vertical asymptotes, x and y intercepts)
Graph the function and show the intercepts and asymptotes:
It is not in our curriculum: no slant graphs
Algebra 2, CPM
Algebra 2
It is not in our curriculum
Algebra 2/Trigonometry
Precalculus
Identify all asymptotes, holes, and intercepts (write as equations of lines) for: . / SFCC – Beginning & Intermediate Algebra
What is the domain of
SCC – Intermediate Algebra
It is not in our curriculum / Algebra Concepts
Find the vertical asymptotes of , then sketch the graph of f.
Precalculus I
Graph the following function

f.Given a graph or graphical features, including degrees, intercepts, asymptotes, and/or holes discontinuities), generate an algebraic representation of a polynomial or rational function.

Core 1
Core 2
Core 3
Integrated 3
It is not in our curriculum
We do this only with linear and quadratic functions.

Algebra 2, CPM
Algebra 2
It is not in our curriculum
Algebra 2/Trigonometry
Precalculus
Given the graph, find the function. Leave the function in factored form.
/ SFCC – Beginning & Intermediate Algebra
SCC – Intermediate Algebra
It is not in our curriculum / Algebra Concepts
Find the symbolic form of a rational function whose graph has vertical asymptotes at and
Precalculus I
Construct a rational function with the graph shown. Justify your answer.

g.Sketch the graph of a polynomial given the degree, zeros, max/min values, and/or initial conditions.

Core 1
Core 2
Core 3
Integrated 3
Algebra 2, CPM
Algebra 2
It is not in our curriculum
Algebra 2/Trigonometry
Precalculus
Find a polynomial function with integer coefficients that has the given properties.
Zeros: (4,0) , (3i,0) , (-3i,0); y-intercept: (0,-9)
Find the polynomial function with leading coefficient 2 that has the given degree and zeros.
Degree 3, with -2,1, and 4 as zeros
Find the polynomial function with leading coefficient 2 that has degree 2 with, -2,1,4 as zeros. / SFCC – Beginning & Intermediate Algebra
SCC – Intermediate Algebra
It is not in our curriculum / Algebra Concepts
Sketch the graph of by first finding the zeros and noting the multiplicity of each.
Precalculus I
Find and sketch the graph of a polynomial of degree 5 with integer coefficients that has zeros , and ; and as ,

h.Graphically/numerically construct new functions using addition, subtraction and composition.

Core 1
Core 2
Core 3
Integrated 3
Algebra 2, CPM
Algebra 2
It is not in our curriculum
Algebra 2/Trigonometry
Course not indicated
Precalculus / SFCC – Beginning & Intermediate Algebra
Let and Find the following:
(a)
(b)
(c)
(d)
SCC – Intermediate Algebra
It is not in our curriculum / Precalculus I
Let and
What is , and and describe there domain and range.
The following graphs represent a company’s costs and revenue. Draw a graph of the profit function. When does the company start to make a profit?



i.Identify the components of composite functions (e.g., given, find suitable functions f, g, and h) and determine the domain and range.

Core 1
Core 2
Core 3
Integrated 3
Algebra 2, CPM
Algebra 2
It is not in our curriculum
Algebra 2/Trigonometry
Precalculus
Find and such that the function can be described as : / SFCC – Beginning & Intermediate Algebra
SCC – Intermediate Algebra
It is not in our curriculum / Precalculus I
If , find and such that

Express where is a piecewise function and .