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1 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 3 12 12 12 12 12 Objective 3: Multiply polynomials Remember laws of exponents: When multiplying monomials, multiply the coefficients 5 3 5 and add the exponents. ( x 4 y 3 )( xy 5 ) x 5 y 8 3 4 4 2 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. 3 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. 4 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 5 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 6 Y=X y X Y = X2 y x 7 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. 8 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. 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. 9 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 Distance = rate · time Sum indicates addition, Difference—subtraction Product--multiplication Quotient—division Consecutive integers are written: X, X+1, X+2, etc.