Download Midterm Review Notes - Spring

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Advanced Algebra II – Midterm Review
Important Formulas and Tips
Chapter 1 – Equations and Inequalities
SOLVING LINEAR EQUATIONS
- Distribute if needed
- Combine like terms
- Isolate your variable and solve
SOLVING LINEAR INEQUALITIES
- Distribute if needed
- Combine like terms
- Isolate your variable
- Remember to flip the inequality sign if you multiply or divide by a negative
- Your solutions are usually graphed on a number line with shading.
SOLVING ABSOLUTE VALUE EQUATIONS
- Isolate the absolute value
- Transform the original equation into two linear equations (one negative and one positive)
- Solve each equation (you should have 2 answers)
SOLVING ABSOLUTE VALUE INEQUALITIES
- Isolate the absolute value
- Transform the inequality into an equivalent inequality
- Less Than becomes an AND written as one 3 part inequality: -c < ax + b < c
-c ≤ ax + b ≤ c
- Greater Than becomes an OR written as 2 inequalities: ax + b > c OR ax + b < -c
ax + b ≥ c OR ax + b ≤ -c
Chapter 2 – Linear Equations and Functions
RELATIONS
- A relation is a function where every input value has exactly one output value
- Domain are the X values, Range are the Y values
- Vertical line test: A relation is a function if and only if no vertical line intersects the graph of the relation
at more than 1 point
LINES
- To find the slope through 2 points: m  y2  y1
x2  x1
- Horizontal lines - have slope of 0 and are the form y  c .
- Vertical lines - have an undefined slope and are in the form x  c .
- Perpendicular lines have slopes that are opposites and reciprocals
- Parallel lines have the same slope
- Slope-intercept form: y  mx  b
- Point-slope form: y  y1  m( x  xz ) - use when given a point and the slope
- To find the x-intercept of a line, plug in 0 for y and solve for x: (#, 0)
- To find the y-intercept of a line, plug in 0 for x and solve for y: (0 , #) or (0, b)
ABSOLUTE VALUE
- Absolute value functions are of the form: y  a | x  h |  k , and makes a “v”-shaped graph.
- Vertex is (h, k), remember to switch the sign of your h
- When graphing, plot your vertex, then use ‘a’ as your slope to plot the next point, use symmetry to get
the other side of the graph.
GRAPHING LINEAR INEQUALITY
- Determine if the line is dashed or solid (depending on the inequality symbol)
- Graph the line using either slope-intercept form or the x- and y-intercepts.
- Test the point (0, 0) if you can to determine where to shade. If the test point is a solution, shade where
the point lies, if it is not, shade where the point is not.
Chapter 3 – Linear Systems
SOLVING LINEAR SYSTEMS
1.) Substitution method:
- Solve one of the equations for one of its variables
- Substitute the expression from step 1 into the other equation and solve for the other variable
- Substitute the value from step 2 into one of the original equations and solve.
2.) Elimination method:
- Multiply one or both of the equations by a constant to obtain coefficients that differ only in sign
for one of the variables
- Add the revised equations from step 1. Combining like terms will eliminate one of the variables.
Solve for the remaining variable
- Substitute the value in step 2 into either of the original equations and solve for the other variable
GRAPHING SYSTEMS OF LINEAR INEQUALITIES
- Graph each inequality in the system
- Identify the region that is common to all the graphs, (where the shading overlaps, darken in this area)
Chapter 4 – Quadratic Functions and Factoring
GRAPHING QUADRATIC FUNCTIONS
Standard form y  ax 2  bx  c
b
, then plug in x to find y
2a
2. Type the function into your graphing calculator to find a table of values. Look for symmetry!
1. To find the vertex: x 
Vertex form y  a( x  h)2  k
1. The vertex is (h, k), remember to flip the sign of h
3. Type the function into your graphing calculator to find a table of values. Look for symmetry!
Intercept form y  a( x  p)( x  q)
1. The x-intercepts are (p, 0) and (q, 0), remember to flip the signs of p and q
pq
2. To find the vertex: x 
then plug in x to find y. (Average the x-intercepts!)
2
Additional Topics:




Axis of symmetry: x-coordinate of the vertex.
Maximum/minimum: corresponds to the y-value of the vertex. If the graph opens up, the vertex
is a minimum, if the graph opens down the vertex is a maximum.
Domain:
(all real numbers)
Range: Dependent on the y-value of the vertex and the orientation of the graph.
4.3 and 4.4 : Factoring Quadratics

When the a value is 1, find the factors of c that add to b and use these factors in your binomials.
Factoring Steps:
1.) Factor any common monomials (GCFs)
2.) If it is a Binomial:
a.) Difference of Squares
a 2  b2  (a  b)(a  b)
Example : 4 x2  9  (2 x  3)(2 x  3)
b.) If not a difference of 2 squares, then you are finished
3.) If it is a Trinomial:
a.) Perfect Square Trinomial
a 2  2ab  b2  (a  b)2
Example : x2  6 x  9  ( x  3)2
OR
a 2  2ab  b2  (a  b)2
Example : 4 x 2  20 x  25  (2 x  5)2
b.) Guess and check using FOIL with the factors and signs. OR
c.) Systematic method using factoring by grouping.
**When solving, set each factor equal to zero and solve for x. **
2.) Square Roots
Vocab:
Conjugates a + √𝑏 and a - √𝑏
Product Property: √𝑎𝑏 = √𝑎 √𝑏, when simplifying, you want the radicand (number under square root
symbol), to have no factors which are a perfect square.
𝑎
Quotient Property: √𝑏 =
√𝑎
√𝑏
, CANNOT HAVE A SQUARE ROOT IN THE DENOMINATOR, YOU MUST
RATIONALIZE. (SIMPLIFY)
Form in denominator
√𝑏
Multiply numerator
and denominator by
√𝑏
a + √𝑏
a - √𝑏
a - √𝑏
a + √𝑏
When solving quadratics using square roots:
1). Isolate the squared term
2). Take the square root of both sides (DON’T FORGET TO PUT ± IN FRONT OF YOUR SQUARE ROOT)
3). Simplify the radical completely
Complex Numbers (a + bi)
√−𝟏 = i, i2 = -1
- When solving quadratics, do as you normally would, isolate the square term, take the square root of
both sides (DON’T FORGET THE ±)
- Write the square root in terms of i
- Simplify
Operations with Complex Numbers
-When adding and subtracting, combine all real numbers, combine all complex numbers, write in
standard form (a + bi)
- When multiplying, simplify by distributing or foiling, write in standard form as combining like terms
(don’t forget to simplify with i 2  1 ).
- When dividing, multiply top and bottom by the complex conjugate (a + bi  a – bi). FOIL or distribute
in the numerator and multiply first and last in the denominator. (don’t forget to simplify with i 2  1 ).
3.) Quadratic Formula and Discriminant
-QUADRATIC FORMULA:
𝑥=
−𝑏±√𝑏 2 −4𝑎𝑐
2𝑎
, use when an equation cannot be factored.
-DISCRIMINANT
b2 – 4ac, the discriminant can determine the number and type of solutions for an equation.
Value of
discriminant
b2 – 4ac > 0
b2 – 4ac = 0
b2 – 4ac < 0
Number and type of
solutions
2 real solutions
One real solution
Two imaginary
solutions
Graph of
Y = ax2 + bx + c
Crosses x – axis 2
times
Cross x-axis 1 time
(at the vertex)
Never crosses the xaxis.
GRAPHING QUADRATIC INEQUALITIES
- Determine whether the parabola is dashed or solid
b
, plot the vertex, use your calculator to find other
2a
points to plot by using your table, connect the points with your dashed or solid parabola curve
- Graph the quadratic by finding the vertex, x 
- Test the point (0,0) if possible. If it is a solution, shade where the test point IS, if the point is not a
solution, shade where the test point is NOT
GRAPHING SYSTEMS OF QUADRATIC INEQUALITIES
- Graph each quadratic separately (see above)
- Shade/darken in the region where they overlap
SOLVING QUADRATIC INEQUALITIES ALGEBRAICALLY
- Write and solve the equation by replacing the inequality with an equal sign. Then solve the equation.
- Write your answer a compound inequality (either an and or an or statement).

If the original inequality sign was less than then the compound inequality is an “AND” statement.

If the original inequality sign was greater than then the compound inequality is an “OR” statement.
*Note: Your conclusion statement should make sense based on the correct order of a number line.
WRITING QUADRATIC MODELS
Vertex form: y  a( x  h)2  k – used when you know the vertex and a point.
- Plug in the given vertex for h & k
- Plug in the point you are given for x and y
- Solve for a
- Write the equation including the values for a, h and k
Intercept form: y  a ( x  p )( x  q ) – used when you know the x-intercepts and a point.
- Plug in the given x-intercepts for p and q
- Plug in the point you are given for x and y
- Solve for a
- Write the equation including the values for p, q and a