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Alabama High School Graduation Exam (AHSGE)
Mathematics Study Tips
Compiled by
Linda Bridges, UAH AMSTI Math Specialist
Standard I—The student will be able to perform basic operations on
algebraic expressions.
Objective 1: Apply order of operations
Remember: Please Excuse MyDear AuntSally.
This is a four-step (not a 6-step) memory technique.
Perform operations in parentheses, simplify expressions with exponents, Multiply and
Divide in order from left to right, Add and Subtract in order from left to right.
Objective 2: Add and Subtract polynomials
You can only add and subtract like terms------for instance, you can add constant terms to
other constant terms, x-terms to other x-terms, y-terms to other y-terms, x 2 terms to
other x 2 terms, xy terms to other xy terms, etc. You add or subtract the coefficients
and leave the exponents in the variable terms exactly as they are---Do Not change the
exponents in the variable portion of the terms. (3x4 + 5x4 = 8x4)
Remember to use the distributive property if necessary to simplify expressions before
adding like terms together.
6a(5a +2) + 14a –10 =
30a2 + 12a +14a – 10=
30a2 + 26a – 10
If terms contain algebraic fractions, remember to get a common denominator before
adding (or subtracting) the terms.
2a  6 5  4a 4(2a  6) (5  4a) 8a  24  5  4a 4a  19

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
3
12
12
12
12
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Objective 3: Multiply polynomials
Remember laws of exponents: When multiplying monomials, multiply the coefficients
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3
5
and add the exponents. ( x 4 y 3 )( xy 5 )   x 5 y 8
3
4
4
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When raising quantities to powers, raise the coefficients to the indicated power and
multiply exponents. (4 y3m5 )3  64 y9 m15
Do not forget the distributive property.
When multiplying two quantities in parentheses together, remember to multiply the first
term in the first parenthesis by every term in the second parenthesis and then come back
and pick up the next term in the first parenthesis and distribute it across the second
parenthesis, etc.
(3x 2  5 x  4)(3x  2)  3x 2 (3x  2)  5 x(3x  2)  4(3x  2) 
9 x3  6 x 2  15 x 2  10 x  12 x  8  9 x3  21x 2  2 x  8
Objective 4: Factor Polynomials
Remember to look for a common monomial factor first. (Take out the gcf from the
constant terms and also take out the gcf from the variable terms). Remember to factor out
variables raised to the smallest power as part of the gcf. For instance, the gcf of
12 x 4 y 6  3xy5 is 3xy5 and should be factored as 3xy5 (4 x3 y  1) .
Remember the difference of two squares: x 2  y 2  ( x  y)( x  y)
To factor a trinomial, attempt to factor as the product of two binomials.
12 x 2  x  6  (3x  2)(4 x  3)
If there are four terms, try to group them two at the time and factor out common terms.
If there is a common parenthesis, then factor out that common parenthesis.
For instance, 15 y 4  10 y 5  24  16 y  5 y 4 (3  2 y)  8(3  2 y)  (5 y 4  8)(3  2 y)
ALWAYS REMEMBER to look for a common monomial factor first and then
factor completely. There may be more than one factoring step required. For instance,
25 y8  225 y 4  25 y 4 ( y 4  9)  25 y 4 ( y 2  3)( y 2  3)
Standard II—The student will be able to solve equations and
inequalities.
Objective 1: Solve multi-step equations of first degree
Steps:
 Simplify each side of the equation by using distributive property and/or
combining like terms, etc.
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
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Move the variable to one side of the equation by adding or subtracting a variable
term to or from both sides of the equation.
Get rid of addition and/or subtraction on the side with the variable by performing
opposite operation to both sides
Get rid of multiplication and/or division on the side with the variable by
performing opposite operation to both sides.
Objective 2: Solve quadratic equations that are factorable.
Get all terms over to one side so that the other side of the equation is 0.
Factor the quadratic expression. Set each factor equal to zero and solve.
Objective 3: Solve systems of two linear equations
If the two equations are graphed, look for the point of intersection. This point will be
your solution to the system of two equations.
If the two equations are given, you many solve for one of the variables in terms of the
other variable and then substitute this expression into the other equation—thus creating
one equation with just one variable.
You may line up the variables and constant terms, one equation underneath the other,
and then multiply one equation by some constant in order to make one of the variable
columns add to zero. Then you will be left with one equation with one variable. Solve
for this variable and substitute this value into the other equation to solve for the other
variable.
Objective 4: Solve multi-step inequalities of first degree.
Follow the same procedures as outlined in Objective 1 of Standard II above. Then
remember if you multiply or divide both sides of the inequality by a negative number,
If  3x  9, then x  3
you must change the inequality sign.
Standard III: The student will be able to apply concepts related to
functions.
Objective 1: Identify functions
When given the graph of a set of ordered pairs, you can use the vertical line test to
determine if the graph is that of a function. If you CAN draw a vertical line connecting
two of the points, the relation is NOT a function. If you CANNOT draw a vertical
line connecting two of the points, the relation IS a function.
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If you are given a set of ordered pairs, a table, or a mapping, for every “x” value, there
must be one and only one “y” value in order to classify the relation as a function.
When substituted into the equation of the function, each ordered pair must make the
equation yield a true statement.
Objective 2: Find the range of functions when given the domain.
Remember: Domain: set of “x” values
Range: set of “y” values
To find the range when given the domain of a function, substitute each member of the
domain into the function rule in place of “x” and then solve for the corresponding value
of “y”. The set of “y” values makes up the range of the function.
If asked in this form: Given: f(x) = 3x2 + 4, what is f(-9)? Substitute -9 in place of x
and evaluate the expression. f(-9) = 3(-9)(-9) + 4 = 3(81) + 4 = 243 + 4 = 247.
Standard IV: The student will be able to apply formulas.
Objective 1: Find the perimeter, circumference, area, and volume of geometric
figures.
Formulas to remember:
 Perimeter: add the lengths of all sides of the geometric figure
 All area and volume formulas will be given to you. For problems involving
circles, use 3.14 as an approximation for  . However, you may leave  in your
answer.
Objective 2: Find the distance, midpoint, or slope of a line segment when given two
points.
Formulas are provided on the reference page. When finding slope, be sure to put the
difference of the “y” values in the numerator and the difference of the “x” values in the
denominator. Also, remember to subtract in the same order each time.
If you have a graph of the line segment given, you can start at the lower of the two points
and count up and over to the next point to find the slope of the line segment. Remember:
If you count up and over to the right, the slope is positive. If you count up and over to
the left, the slope is negative.
When finding the distance between two points, you will be expected to simplify radicals.
Think of the largest perfect square factor of the number underneath the square root
symbol and write the square root as the product of that factor and another factor. Then
take the square root of the perfect square factor and place this number outside the square
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root symbol and then leave the non-perfect square factor underneath the square root
symbol. For instance, 104  4  26  2 26
Standard V: The student will be able to apply graphing techniques.
Objective 1: Graph or identify graphs of linear equations
Objective 4: Identify graphs of common relations
Remember, you can always check to see if an equation and a graph correspond by
checking ordered pairs. All ordered pairs which satisfy the equation should lie on the
graph and all points which lie on the graph should satisfy the equation.
Y= mx + b is the slope-intercept form of the equation of a line. The slope is
represented by m and the line crosses the y-axis at b.
Some basic graphs are given below.
Y = “a constant” will feature a horizontal line. In the graph above, y = 3.
Y
X =”a constant” will feature a vertical line. In the graph above, x = -4
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Y=X
y
X
Y = X2
y x
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Objective 2: Graph lines given certain conditions.
X-intercept is where the graph crosses the x-axis. Y-intercept is where the graph crosses
the y-axis.
Y= mx + b is the slope-intercept form of the equation of a line. The slope is
represented by m and the line crosses the y-axis at b.
If you have a graph of the line segment given, you can start at the lower of the two points
and count up and over to the next point to find the slope of the line segment. Remember:
If you count up and over to the right, the slope is positive. If you count up and over to
the left, the slope is negative.
Objective 3: Determine solution sets of inequalities
“Or” represents the union of two graphs. “And” represents the intersection of two
graphs. If the inequality symbol has the equal bar underneath it (≤ or ≥), then you color
in the endpoint. If the inequality is strictly less than (<) or greater than (>), then you
leave an open circle around the endpoint. “Less than” shades to the left and “greater
than” shades to the right.
Standard VI: The student will be able to represent problem situations.
Objective 1: Translate verbal or symbolic information into algebraic expressions or
identify equations or inequalities that represent graphs or problem situations.
To match a graph with an equation, you can test the points to see if the ordered pairs
lying on the graph make a true statement when substituted into the equation. If so, the
graph and the equation match.
To find the equation of a line passing through two points, first find the slope of the line
using the slope formula provided on your reference page. Then write the equation as
y  slope ( x)  b . Substitute one of the ordered pairs into the equation, making sure you
substitute the x value in place of x and the y value in place of y. Then solve for “b”.
Write the equation in y = mx + b form.
Read each word carefully when translating verbal information into algebraic expressions.
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Standard VII: The student will be able to solve problems involving a
variety of algebraic and geometric concepts.
Objective 1: Apply properties of angles and relationships between angles.
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Vertical angles are congruent.
The sum of the measures of two supplementary angles is 180 degrees.
The sum of the measures of two complementary angles is 90 degrees.
If two angles form a straight line, the sum of their measures is 180 degrees.
If two parallel lines are cut by a transversal, alternate interior angles are
congruent, corresponding angles are congruent, and same side interior angles are
supplementary.
The sum of the measures of the interior angles of a convex polygon is 180(n-2),
where n is the number of sides of the polygon.
The sum of the measures of the interior angles of a triangle is 180 degrees.
Objective 2: Apply Pythagorean Theorem
Given a right triangle with legs of lengths a and b and hypotenuse of length c, then the
a 2  b2  c2
following relationship is always true.
Objective 3: Apply properties of similar polygons
If two polygons are similar, their corresponding angles are congruent and the lengths of
corresponding sides are in proportion.
Objective 4: Apply properties of plane and solid geometric figures.
Use the reference page to find area and volume formulas.
Objective 5: Determine measures of central tendency
Mean—arithmetic average (Add all the numbers in the set and divide by the number of
items in the set)
Median—middle First of all, arrange numbers in order from smallest to largest. If
there is an odd number of data observations, the median will be the middle number in the
ordered list. If there is an even number of observations in the data set, the median will be
the average of the two middle values.
Mode—the observation which occurs most often. There may be no mode, one mode, or
more than one mode for a given set of data.
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Objective 6: Determine probabilities
To find the probability of two events connected with “and”, multiply the two separate
probabilities. For instance, the probability of drawing two cards from a standard deck of
playing cards and the first one is an ace and the second one is a king is (4/52) (4/51) =
4/663
When finding the probability of two mutually exclusive events A or B, add the separate
probabilities. For instance, the probability of tossing a number cube and getting a “5” or
a “6” is 1/6 + 1/6 = 2/6 or 1/3.
If two events are not mutually exclusive, then the probability of A or B is :
P(A) + P(B) – P(A and B). For instance, the probability of drawing a card from a
standard deck of cards and getting a “4” or a “red card” is.
P(4) + P(red) – P(4 and red) = 4/52 + 26/52 – 2/52 = 28/52 or 7/13.
Objective 7: Solve problems involving direct variation
If y varies directly as x, then set up the equation: y = kx. Substitute values into x and y
and solve for k. Then rewrite the equation and substitute in for one variable and solve for
the other variable.
For example:
Y varies directly as X. If X = 6 when Y = 16, then find the value of X when Y = 64.
Y=kx
16 = k (6)
8/3 = k
Y = 8/3 X
64 = 8/3 x
24 = X
Objective 8: Solve problems involving algebraic concepts
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Distance = rate · time
Sum indicates addition, Difference—subtraction Product--multiplication
Quotient—division
Consecutive integers are written: X, X+1, X+2, etc.