The sat skill exam 7

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5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 122 – THE SAT MATH SECTION – Example: I SOLATING VARIABLES U SING F RACTIONS It may be necessary to use factoring in order to isolate (3x + 1)(7x + 10) a variable in an equation. 3x and 7x are the ﬁrst pair of terms, 3x and 10 are the outermost pair of terms, Example: 1 and 7x are the innermost pair of terms, and If ax – c = bx + d, what is x in terms of a, b, c, 1 and 10 are the last pair of terms. and d? Therefore, (3x)(7x) + (3x)(10) + (1)(7x) + (1)(10) = 21x2 + 30x + 7x + 10. 1. The ﬁrst step is to get the x terms on the same After we combine like terms, we are left with side of the equation. the answer: 21x2 + 37x + 10. ax – bx = c + d 2. Now you can factor out the common x term on Factoring the left side. Factoring is the reverse of multiplication: x(a – b) = c + d 3. To ﬁnish, divide both sides by a – b to isolate the 2(x + y) = 2x + 2y Multiplication variable of x. 2x + 2y = 2(x + y) Factoring x(a – b) c+d = a–b a–b T HREE B ASIC T YPES FACTORING 4. The a – b binomial cancels out on the left, result- OF Factoring out a common monomial: ■ ing in the answer: 10x2 – 5x = 5x(2x – 1) and xy – zy = y(x – z) c+d x= a–b Factoring a quadratic trinomial using the reverse ■ of FOIL: Q uadratic Trinomials y2 – y – 12 = (y – 4) (y – +3) and A quadratic trinomial contains an x2 term as well as an 2 – 2z + 1 = (z – 1)(z – 1) = (z – 1)2 z x term. x2 – 5x + 6 is an example of a quadratic trino- Factoring the difference between two squares ■ mial. It can be factored by reversing the FOIL method. using the rule: a2 – b2 = (a + b)(a – b) and Start by looking at the last term in the trinomial, ■ x2 – 25 = (x + 5)(x – 5) the number 6. Ask yourself, “What two integers, when multiplied together, have a product of posi- R EMOVING C OMMON FACTOR A tive 6?” If a polynomial contains terms that have common fac- Make a mental list of these integers: ■ tors, the polynomial can be factored by using the 1 × 6 –1 × –6 2 × 3 –2 × –3 reverse of the distributive law. Next, look at the middle term of the trinomial, in ■ this case, the –5x. Choose the two factors from Example: the above list that also add up to –5. Those two In the binomial 49x3 + 21x, 7x is the greatest factors are: common factor of both terms. –2 and –3 Thus, the trinomial x2 – 5x + 6 can be factored as Therefore, you can divide 49x3 + 21x by 7x to ■ (x – 3)(x – 2). get the other factor. 49x3 + 21x 49x3 21x = 7x2 + 3 = + 7x 7x 7x Thus, factoring 49x3 + 21x results in 7x(7x2 + 3). 122
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 123 – THE SAT MATH SECTION – Be sure to use the FOIL method to double-check Using the zero-product rule, it can be deter- ■ your answer: mined that either x + 5 = 0 or that x – 3 = 0. (x – 3)(x – 2) = x2 – 5x + 6 x+5= 0 or x–3= 0 The answer is correct. –5 –5 + 3 +3 x = –5 or x = +3 Algebraic Fractions Algebraic fractions are very similar to fractions in Thus, the possible values of x are –5 and 3. arithmetic. Example: Solving Quadratic Equations by Factoring x x Write 5 – 10 as a single fraction. ■ If a quadratic equation is not equal to zero, you Just like in arithmetic, you need to ﬁnd the LCD need to rewrite it. of 5 and 10, which is 10. Then change each frac- Example: Given x2 – 5x = 14, you will need to subtract 14 tion into an equivalent fraction that has 10 as a from both sides to form x2 – 5x – 14 = 0. This denominator. quadratic equation can now be factored using x x x(2) x – = 5(2) – 10 the zero-product rule. 5 10 2x x ■ It may be necessary to factor a quadratic equation = 10 – 10 x before solving it and to use the zero-product rule. = 10 Example: x2 + 4x = 0 must ﬁrst be factored before it can Reciprocal Rules be solved: x(x + 4). There are special rules for the sum and difference of reciprocals. Memorizing this formula might make you Graphs of Quadratic Equations more efﬁcient when taking the SAT. The (x,y) solutions to quadratic equations can be plot- 1 x+y ted. It is important to look at the equation at hand If x and y are not 0, then x + 1y = ■ xy and to be able to understand the calculations that are 1 1 y–x If x and y are not 0, then x – y = ■ xy being performed on every value that gets substituted into the equation. Q uadratic Equations For example, below is the graph of y = x2. A quadratic equation is an equation in which the greatest exponent of the variable is 2, as in x2 + 2x – 15 5 = 0. A quadratic equation has two roots, which can be 4 found by breaking down the quadratic equation into 3 two simple equations. 2 1 x Zero-Product Rule –7 –6 –5 –4 –3 –2 –1 1234567 –1 The zero-product rule states that if the product of two or –2 more numbers is 0, then at least one of the numbers is 0. –3 y Example: (x + 5)(x – 3) = 0 123
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 124 – THE SAT MATH SECTION – For every number you put into the equation (as Rational Equations an x value), you know that you will simply square the and Inequalities Recall that rational numbers are all numbers that can number to get the corresponding y value. 2 be written as fractions ( 3 ), terminating decimals (.75), The SAT may ask you to compare the graph of y = x2 with the graph of y = (x – 1)2. Think about what hap- and repeating decimals (.666 . . . ). Keeping this in mind, it’s no surprise that rational equations are just pens when you put numbers (your x values) into this equations in fraction form. Rational inequalities are equation. If you have an x = 2, the number that gets also in fraction form and involve the symbols , ≤, squared is 1. The graph will look identical to the and ≥ instead of an equals sign. y = x2 graph, except it will be shifted to the right by 1: Example: 5 (x + 5)(x2 – x – 12) Given = 10, ﬁnd the value of x. 4 x2 + x – 20 3 Factor the top and bottom: 2 1 (x + 5)(x + 3)(x – 4) = 10 (x + 5)(x – 4) x –7 –6 –5 –4 –3 –2 –1 1234567 –1 Note that you can cancel out the (x + 5) and the –2 (x – 4) terms from the top and bottom to yield: –3 x + 3 = 10 y Thus, x = 7. How would the graph of y = x2 compare with the graph of y = x2 – 1? Radical Equations In this case, you are still squaring your x value, Some algebraic equations on the SAT will include the and then subtracting 1. This means that the whole square root of the unknown. The ﬁrst step is to isolate graph of y = x2 has been moved down 1 point. the radical. When you have accomplished this, you can then square both sides of the equation to solve for the 5 unknown. 4 3 Example: 2 4 b + 11 = 27 1 x –7 –6 –5 –4 –3 –2 –1 1234567 To isolate the variable, subtract 11 from both –1 sides: –2 4 b = 16 –3 y Next, divide both sides by 4: b=4 Last, square both sides: b2 = 42 b = 16 124
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 125 – THE SAT MATH SECTION – S equences Involving Substitution Exponential Growth Substitution involves solving for one variable in terms When analyzing a sequence, you always want to try of another and then substituting that expression into and ﬁnd the mathematical operation that you can per- the second equation. form to get the next number in the sequence. Look carefully at the sequence: Example: 2p + q = 11 and p + 2q = 13 2, 6, 18, 54, . . . 1. First, choose an equation and rewrite it, isolating one variable in terms of the other. It does not You probably noticed that each successive term is matter which variable you choose. found by multiplying the prior term by 3; (2 × 3 = 6, 2p + q = 11 becomes q = 11 – 2p 6 × 3 = 18, and so on.) Since we are multiplying each 2. Second, substitute 11 – 2p for q in the other term by a constant number, there is a constant ratio equation and solve: between the terms. Sequences that have a constant p + 2(11 – 2p) = 13 ratio between terms are called geometric sequences. p + 22 – 4p = 13 On the SAT, you may, for example, be asked to 22 – 3p = 13 ﬁnd the thirtieth term of a geometric sequence like the 22 = 13 + 3p one above. There is not enough time for you to actu- 9 = 3p ally write out all the terms, so you should notice the p=3 pattern: 3. Now substitute this answer into either original equation for p to ﬁnd q. 2, 6, 18, 36, . . . 2p + q = 11 Term 1 = 2 2(3) + q = 11 Term 2 = 6, which is 2 × 3 6 + q = 11 Term 3 = 18, which is 2 × 3 × 3 q=5 Term 4 = 54, which is 2 × 3 × 3 × 3 4. Thus, p = 3 and q = 5. Another way of looking at this, would be to use Linear Combination exponents: Linear combination involves writing one equation over another and then adding or subtracting the like terms Term 1 = 2 so that one letter is eliminated. Term 2 = 2 × 31 Term 3 = 2 × 32 Example: Term 4 = 2 × 33 x – 9 = 2y and x + 3 = 5y 1. Rewrite each equation in the same form. So, if the SAT asks you for the thirtieth term, you x – 9 = 2y becomes x – 2y = 9 and x + 3 = 5y know that term 30 = 2 × 329. becomes x – 5y = 3 Systems of Equations A system of equations is a set of two or more equations with the same solution. Two methods for solving a system of equations are substitution and linear combination. 125
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 126 – THE SAT MATH SECTION – 2. If you subtract the two equations, the x terms All of the solutions to f(x) are collectively called will be eliminated, leaving only one variable: the range. Values that f(x) cannot equal are said to be Subtract: outside of the range. x – 2y = 9 The x values are known as the independent vari- –(x – 5y = 3) ables. The outcome of the function depends on the x val- 3y 6 ues, so the y values are called the dependent variables. = 3 3 y = 2 is the answer. Qualitative Behavior of Graphs 3. Substitute 2 for y in one of the original equations and Functions and solve for x: In addition to being able to solve for f(x) and make x – 2y = 9 judgments regarding the range and domain, you should x – 2(2) = 9 also be able to analyze the graph of a function and inter- x–4=9 pret, qualitatively, something about the function itself. x–4+4=9+4 Look at the x-axis, and see what value for f(x) cor- x = 13 responds to each x value. 4. The answer to the system of equations is y = 2 For example, consider the portion of the graph and x = 13. shown below. For how many values does f(x) = 3? Functions, Domain, and Range 5 Functions are written in the form beginning with: 4 3 f(x) = 2 1 For example, consider the function f(x) = 3x – 8. x If you are asked to ﬁnd f(5), you simply substitute the –7 –6 –5 –4 –3 –2 –1 1234567 –1 5 into the given function equation. –2 –3 f(x) = 3x – 8 y becomes f(5) = 3(5) – 8 When f(x) = 3, the y value (use the y-axis) will f(5) = 15 – 8 = 7 equal 3. As shown below, there are ﬁve such points. In order to be classiﬁed as a function, the function in question must pass the vertical line test. The verti- 5 cal line test simply means that a vertical line drawn 4 through a graph of the function in question CANNOT 3 pass through more than one point of the graph. If the 2 1 function in question passes this test, then it is indeed a x function. If it fails the vertical line test, then it is NOT a –7 –6 –5 –4 –3 –2 –1 1234567 –1 function. –2 All of the x values of a function, collectively, are –3 called its domain. Sometimes, there are x values that are y outside of the domain, and these are the x values for which the function is not deﬁned. 126
5658 SAT2006[04](fin).qx 11/21/05 6:44 PM Page 127 – THE SAT MATH SECTION – G eometr y Review To begin this section, it is helpful to become familiar with the vocabulary used in geometry. The list below deﬁnes some of the main geometrical terms. It is followed by an overview of geometrical equations and ﬁgures. arc part of a circumference area the space inside a two-dimensional ﬁgure bisect cut in two equal parts circumference the distance around a circle chord a line segment that goes through a circle, with its endpoint on the circle identical in shape and size. The geometric notation of “congruent” is ≅ . congruent diameter a chord that goes directly through the center of a circle—the longest line you can draw in a circle equidistant exactly in the middle hypotenuse the longest leg of a right triangle, always opposite the right angle line a straight path that continues inﬁnitely in two directions. The geometric notation for a line is AB. line segment the part of a line between (and including) two points. The geometric notation for a line segment is PQ. parallel lines in the same plane that will never intersect perimeter the distance around a ﬁgure perpendicular two lines that intersect to form 90-degree angles quadrilateral any four-sided ﬁgure radius a line from the center of a circle to a point on the circle (half of the diameter) ray a line with an endpoint that continues inﬁnitely in one direction. The geometric notation for a ray is AB . tangent line a line meeting a smooth curve (such as a circle) at a single point without cutting across the curve. Note that a line tangent to a circle at point P will always be perpendicular to the radius drawn to point P. volume the space inside a three-dimensional ﬁgure 127