A2A MAT115 College Algebra Content Topics for Exam 3
1. Functions
a. Determine if a graph or set of points is a function (vertical line test)
b. Be able to explain in sentence form why an equation or graph is or is not a function
c. Be able to write an equation in function notation.
d. Determine the domain of a function.
i. Find the values for which a rational function is undefined and when a square root function is
undefined.
ii. Find the domain and range of a continuous graph, a graph of points, ordered pairs.
iii. Be able to write domain in interval notation.
e. Given a graph, determine when f(x) has a positive or negative function value and find any relative
maximum or minimum points.
f.
Find f(a) and f(x) = b, from a continuous or graph of points.
g. Given a function find f(a), f(a + h),
h. Be able to find the average rate of change of a function
i.
f ( x  h)  f ( x )
h
Find f(a) and f(x) = b, from a continuous or graph of points.
2. Symmetry
a. Show algebraically whether or not a function is even, odd or neither
b. Determine even or odd symmetry based on the look of a graph.
3. Transformations - be able to apply the following transformations to the basic graphs below.
Know the basic graphs of the functions y  x , y  x 2 , y  x3 , y  x , y 
a. Horizontal shift – Examples  ( x  a) 2 ,
3
b. Vertical shift – Examples  x  a ,
x, y
1
x
x  a ...
x a …
c. Vertical reflection – Examples   f (x)   (x)2 ,  x , 
1
...
x
1
, ( x) 2 ,  x …
x
e. Vertical stretching – Examples  k  f (x) where k  1  2( x) 2 , 3 x , 5 x , 2( x)3 …
d. Horizontal reflection – Examples  f ( x) 
f.
Multiple combinations   5( x  3) 2 , 2 x  6  5 , 3(x  5)3  5 …
4. Piecewise functions
a. Given a piecewise graph
i.
Be able to evaluate a piecewise graph for a given value  find f(2)…
ii.
Be able to find the domain and range of a piecewise graph
iii.
Be able to find the piecewise function
b. Given a Piecewise equation
i.
Be able to evaluate a piecewise graph for a given value  find f(2)…
ii.
Be able to graph a piecewise function
5. General function graphs – be able to find the following for any of the functions we have worked with. You
may want to make a list of the different types of functions you have seen.
a. Find the intercepts
b. Find the domain and range
c. Be able to identify the local and absolute extrema of the graph
d. Determine intervals of increase and decrease
6. Variation: Be able to represent direct and inverse variation models
7. Domain - Identify which functions have domain restrictions and what those restrictions are (look for square
roots and rational (fraction) expressions)
8. Combining functions
a. Be able to add, subtract, divide and multiply functions in terms of x  ( f  g )( x) = answer in terms of x
b. Be able to evaluate combined functions given a value  ( f  g )(3) = number
c. Be able to find the domain of combined functions
d. Be able to combine functions given a graph
9. Composition of functions
a. Be able to find a composition of functions in terms of x  f ( g ( x)) = answer in terms of x
b. Be able to evaluate a composition of functions  f (g (2)) = a number
10. Quadratic Functions
11. Given the graph of a quadratic functions
a. Find the x- and y-intercepts, vertex, axis of symmetry, concavity, domain
12. Given a quadratic function as an equation
a. Find the concavity, x- and y-intercepts, vertex and axis of symmetry and sketch the graph.
b. Be able to find the vertex by completing the square or using the vertex theorem
13. Quadratic word problems involving maximizing and minimizing (disguised questions asking for the vertex)
14. Word problems involving the path an object takes in terms of height and time