Ms.
Liefland Algebra
2/Trig Semester Review – Test Wednesday
Chapter
1
- Looking at a table to tell if a
relationship is linear
- Writing equations of lines
- (Solving absolute-value equations
and knowing their graphs)
- Finding domain and range of
equations
Chapter
2
- Finding the inverse of a function
- Reverse x and y
- Re-solve for y
- Operations with functions (i.e. f+g,
fog)
- Properties of Exponents
- Piecewise functions
- Transformations of graphs and
functions
Chapter
3
- System of equations (vocab 3.1)
- Solving systems using substitution,
elimination
- Graphing linear inequality
- Graphing system of linear equations
- Parametric equations and
“eliminating the parameter” (3.6)
Chapter
5
- Writing standard form of a quadratic
equation
- Finding the vertex of a quadratic
function, knowing whether it opens up or down
- Solving quadratic equations using
inverse operations (step by step)
- Factoring quadratic expressions and
using zero-product property to find zeros = set factors equal to zero
- Writing a function in vertex form
- Using quadratic formula
- Using the discriminant to determine
the number of real roots
- The part of the quadratic formula
under the square root
- If it is positive: two real roots
- If it is negative: two imaginary
solutions
- If it is zero: you have a double
root
- Solving quadratic equations with
imaginary solutions
- Using operations with imaginary
numbers
- Writing a quadratic equation from
three points (system of equations for coefficients)
Chapter
6
- Exponential growth or decay?
- Changing between logarithmic form
and exponential form
- Condensing/expanding logarithmic
equations
- Solving log equations
- Using the natural logarithm base “e”
Chapter
7
- Classify the polynomial by degree
- End behavior (arrows)
- Finding zeros of a polynomial =
knowing that they are x-intercepts
- Synthetic division
- Knowing that when you divide by a
factor and get remainder 0, then it is a factor. Otherwise, it is not a factor
Chapter
8
- Finding vertical asymptotes and
horizontal asymptotes of rational equations
Chapter 1 Review – Linear Representations
Page 77: 1, 2, 8, 10, 12, 25
#1: not linear – quadratic!! – the y terms are squares
#2: linear – x and y have a constant change
#8: y = 2x + 6 (use
point-slope form first) y-y1 = m(x-x1) with point(x1,y1)
#10: find slope first à y = -12x + 41
#12: y = (3/2)x + (33/2) – perpendicular lines have NEGATIVE
RECIPROCALS
#25: x>1 and x<4 – graph the intersection point with “and”
– between 1 and 4
Chapter 2 Review – Numbers and Functions
Page 147: 10, 15, 19, 22, 25, 28
#10: 3 / (r * s3
*t) - take all of the terms with a
negative exponent and move them all DOWNSTAIRS
#15: Not a function!!!
Same x leads to more than one y-value – this would violate the vertical line
test
#19: = 3x – 2
#22: (g o f)(3) = 4 I
write this instead as g(f(3)) à evaluate f(3) then
plug to g
- Take 3 and
plug it in to f, and something comes out
- Take that
answer and plug it in to g, and you get the answer 4
#25: reverse all of the coordinates to find the inverse, YES it
is a function
#28: the inverse is
plus/minus the 4th root of (x+3); this is not a function
- Solving for
the Inverse: Switch x and y. Solve for y.
- NOTE: how to
tell if a graph is the graph of a function: vertical line test
- How to tell
if the INVERSE of a graph is a function: horizontal line test
Chapter 3 Review – Systems of Linear Equations and Inequalities
Page 207: 3, 8, 9, 19
#3: using substitution, you get intersection point (5,3)
#8: has no solution!!!
#9: from elimination, you get intersection point (3, -4)
#19: Eliminate the parameter!! – follow the steps below or solve
both for t and set these equal
·
Steps for eliminating the parameter in #19 (essentially
substitution)
1. Your equations are x =
t - 4 and y = 2t - 11
2. Solve one of them for
t.
So t = x+4
3. Plug that into the
other equation
so y=2(x+4) – 11
4. Simplify into your
answer…. Y = 2x – 3
Chapter
5 Review – Quadratic Equations
Page
345: 2, 4, 5, 9-12 all, 18, 22, 23, 25,
[[30 Trig Only)]]
2. a=-5 , b=-30, c=35
4.
+/- 3rt(3) or +/-5.2
5.
7 +/- 2rt(3)
9. 0 and 9
10. 4 and -4
11. ½ double root à remember how this looks GRAPHICALLY
12. -2 and 5
18. 1 or -5
22. 0 real solutions – REMEMBER
imaginary solutions COME IN PAIRS
23. 2
25. -4 + 4i
30. create three equations by using
a coordinate point as x and y for three equations in standard form
then solve for the coefficients à
solve the system of equations
Chapter
6 Review – Exponential and Logarithmic Functions
Page
415: 3, 4, 6, 7, 15, 16, 17, 20, 21, 24
3. growth because (1.01) is the
“multiplier” and it is bigger than 1
4. growth for the same reason, if it
was less than one things would be getting smaller and it would be decay
6. decay because (.25) is the
multiplier and its less than one
7. taking
out interest from cumulative test
15. v=7 because 7 to
the third power is 343
16. v=3 because 9 to
third power is 729
17. v=7776 because 6
to the 5th power is 7776
20. 32
21. 2 because 6 to the 2nd power is 36
or change of base formula
24. 5404.7
because 4 to the power of 6.2 is that answer
Chapter
7 Review - Polynomials
Page
471: 8, 9, 16, 21, 22, 27
8. up; up = end behavior depends on
the DEGREE and whether the leading coefficient is positive or negative
9. up on left; down on right
16. (5x^2)(x-6)(x+6)
21. (x+3) is what is
left over…on the test you will be able to use synthetic div.
22. –x^2 + 3x -2 is what is left over after dividing
27. variable
substitution = solve like regular generic rectangle
but use x-squared instead of x in your factors
so your factors are (x^2 – 4) and (x^2 –
2) and set each factor equal to zero
so your solutions are plus/minus 2 and plus/minus rt. 2
Chapter 7 –
Polynomials Notes
Ø
“The degree” of the polynomial = the highest
exponent that you see
Ø
The number of solutions = the degree of the
polynomial
Ø
Just like with quadratics, we SET THE FACTORS EQUAL TO ZERO to get
solutions.
Ø
Step “Zero” with factoring should always be to
SCAN and see if there is something you can pull out of EVERYTHING – that
becomes a factor
REMEMBER: Imaginary
solutions COME IN PAIRS.
For example a 2nd degree (parabola) can have 2
real or 2 imaginary solutions.
A 3rd degree function can have either 3 real solutions, or it can
have 1 real solution and 2 imaginary solutions.
·
List the possibilities of real/imaginary combos
for a 5th degree function:
Ways to Factor When
the Degree is Two (i.e. quadratic or parabola):
·
Regular “generic rectangle” factoring
·
Quadratic Formula (especially if you suspect
there are IMAGINARY solutions)
·
Difference of Squares
·
Factor by grouping
Ways to Factor When
the Degree is More Than Two:
Every time you divide a polynomial by
a factor, the degree decreases by one. If you can “knock” it down into a
quadratic, you can always get the last two solutions (see above)
·
Step “zero”: Scan to see if there is something
you can pull out of everything. That becomes a factor.
·
Factor by grouping
·
Sum/Difference of Squares
·
“Variable Substitution”
·
Use the rational roots theorem to get a “pile”
of possible rational solutions
·
You can always use a graphing calculator to FIND YOUR SOLUTIONS FOR YOU (real ones):
o
Graph the function in the y=
window
o
Press 2nd button, then CALC/trace
button
o
Go to “zeros”
o
Go to the right of the function, press enter. Go
to the left of the solution, press enter. Press enter one more time.
o
The solution (x-intercept) appears along the
bottom of the screen.
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