Passport to Advanced Math
Example 12: If t is a solution to the equation above and t > 0, what is the value of t?
12(t + 3) = 8(t + 1)(t + 3). Expanding all the products and moving all the terms to the right-hand side gives 0 = t2 − 25. Therefore, the solutions to the equation are t = 5 and t = −5. Since t > 0, the value of t is 5.
Relationships Between Algebraic and Graphical Representations of Functions
A function f(x) has a graph in the xy-plane, which is the graph of the equa tion y = f(x) (or, equivalently, consists of all ordered pairs (x, f(x)). Some questions in Passport to Advanced Math assess your ability to relate prop erties of the function f to properties of its graph, and vice versa. You may be required to apply some of the following relationships: Intercepts. The x-intercepts of the graph of f correspond to values of x such that f(x) = 0; if the function f has no zeros, its graph has no x-intercepts, and vice versa. The y-intercept of the graph of f corresponds to the value of f(0). If x = 0 is not in the domain of f, the graph of f has no y-intercept, and vice versa. Domain and range. The domain of f is the set of all x for which f(x) is defined. The range of f is the set of all y with y = f(x) for some value of x in the domain. The domain and range can be found from the graph of f as the set of all x-coordinates and y-coordinates, respectively, of points on the graph. Maximum and minimum values. The maximum and minimum values of f can be found by locating the highest and the lowest points on the graph, respectively. For example, suppose P is the highest point on the graph of Then the y-coordinate of P is the maximum value of f, and the x-coordinate of P is where f takes on its maximum value. Increasing and decreasing. The graph of f shows the intervals over which the function f is increasing and decreasing. End behavior. The graph of f can indicate if f(x) increases or decreases without limit as x gets very large and positive or very large and negative. Asymptotes. If the values of f approach a fixed value, say K, as x gets very large and positive or very large and negative, the graph of f has a horizontal asymptote at y = K. If f is a rational function whose denominator is zero and numerator is nonzero at x = a, then the graph of f has a vertical asymptote at x = a. Symmetry. If the graph of f is symmetric about the y-axis, then f is an even function, that is, f(−x) = f(x) for all x in the domain of f. If the graph of f is symmetric about the origin, then f is an odd function, that is, f(−x) = −f(x) for all x in the domain of f. Transformations. For a graph of a function f, a change of the form f(x) + a will result in a vertical shift of a units and a change of the form f(x + a) will result in a horizontal shift of a units.
Example 8: A researcher estimates that the population of a city is declining at an annual rate of 0.6%. If the current population of the city is 80,000, which of the following expressions appropriately models the population of the city t years from now according to the researcher's estimate?
According to the researcher's estimate, the population is decreasing by 0.6% each year. Since 0.6% is equal to 0.006, after the first year, the population is 80,000 − 0.006(80,000) = 80,000(1 − 0.006). After the second year, the pop ulation is 80,000(1 − 0.006) − 0.006(80,000)(1 − 0.006) = 80,000(1 − 0.006)2. Similarly, after t years, the population will be 80,000(1 − 0.006)t according to the researcher's estimate. This is choice A.
Exponential Functions, Equations, and Expressions and Radicals
As discussed in Chapter 20, exponential func tions model situations in which a quantity is multiplied by a constant factor for each time period. An exponential function can be increasing with time, in which case it models exponential growth, or it can be decreasing with time, in which case it models exponential decay.
Example 15: The function f(x) = x^4− 2.4x^2is graphed in the xy-plane as shown above. If k is a constant such that the equation f(x) = k has 4 solutions, which of the following could be the value of k?
Choice C is correct. The equation f(x) = k will have 4 solutions if and only if the graph of the horizontal line with equation y = k intersects the graph of f at 4 points. The graph shows that of the given choices, only for choice C, −1, does the graph of y = −1 intersect the graph of f at 4 points.
Example 19: A store manager estimates that if a video game is sold at a price of p dollars, the store will have weekly revenue, in dollars, of r(p) = −4p^2+ 200p from the sale of the video game. Which of the following equivalent forms of r(p) shows, as constants or coefficients, the maximum possible weekly revenue and the price that results in the maximum revenue?
Choice D is correct. The graph of r in the coordinate plane is a parabola that opens downward. The maximum value of revenue corresponds to the vertex of the parabola.
Example 4: A car is traveling at x feet per second. The driver sees a red light ahead, and after 1.5 seconds reaction time, the driver applies the brake. After the break is applied, the car takes x/24 seconds to stop,during which 24 165 feet from the time the driver saw the red light to the time it comes to a complete stop, which of the following equations can be used to find the value of x? time the average speed of the car is x/2 feet per second. If the car travels 165 feet from the time the driver saw the red light to the time it comes to a complete stop, which of the following equations can be used to find the value of x?
During the 1.5-second reaction time, the car is still traveling at x feet per second, so it travels a total of 1.5x feet. The average speed of the car during the interval the x -second braking interval is x feet per second, so over this interval, the 24 2 2 car travels (x )( x )= x feet. Since the total distance the car travels from the 2 24 48 time the driver saw the red light to the time it comes to a complete stop is 165 feet, you have the equation _x2 + 1.5x = 165. This quadratic equation can 48 be rewritten in standard form by subtracting 165 from each side and then multiplying each side by 48, giving x2 + 72x − 7,920, which is choice D.
Analyzing More Complex Equations in Context
Equations and functions that describe a real-life context can be complex. Often it is not possible to analyze them as completely as you can analyze a linear equation or function. You still can acquire key information about the context by analyzing the equation or function that describes it. Questions on the Passport to Advanced Math section may ask you to use an equation describing a context to determine how a change in one quantity affects another quantity. You may also be asked to manipulate an equation to isolate a quantity of interest on one side of the equation. You may be asked to produce or identify a form of an equation that reveals new information about the context it represents or about the graphical representation of the equation.
Example 3: y^5− 2y^4− cxy + 6x In the polynomial above, c is a constant. If the polynomial is divisible by y − 2, what is the value of c?
If this entire expression is divisible by y − 2, then −cxy + 6x must be divisible by y − 2. Thus, −cxy+6x=(y−2)(−cx)=−cxy+2cx.Therefore,2c=6, and the value of c is 3
Example 7: What is the sum of the solutions of (2x − 1)^2 = (x + 2)^2?
Ifaandbarerealnumbersanda2 =b2,theneithera=bora=−b.Since (2x − 1)2 = (x + 2)2, either 2x − 1 = x + 2 or 2x − 1 = −(x + 2). In the first case, 1 x = 3, and in the second case, 3x = −1, or x = −_. Therefore, the sum of the solutions x of (2x−1)2 =(x+2)2 is3+ −_ = _ . ( 3) (3) (3,18)
Example: 11: When 6x2 − 5x + 4 is divided by 3x + 2, the result is 2x − 3 R/(3x + 2) where R is a constant. What is the value of R?
Perform long division. The remainder is 10.
Quadratic Functions and Equations
Questions in Passport to Advanced Math may require you to build a quadratic function or an equation to represent a context.
Systems of Equations
Questions on the SAT Math Test may ask you to solve a system of equations in two variables in which one equation is linear and the other equation is quadratic or another nonlinear equation.
Operations with Polynomials and Rewriting Expressions
Questions on the SAT Math Test may assess your ability to add, subtract, and multiply polynomials.
Dividing Polynomials by a Linear Expression and Solving Rational Equations
Questions on the SAT Math Test may assess your ability to work with ratio nal expressions, including fractions with a variable in the denominator. This may include long division of a polynomial by a linear expression or finding the solution to a rational equation.
Example 2: Which of the following is equivalent to 16s^4−4t^2?
Refactor and it can be choice C
Example 10: x − 12 = √x + 44 What is the solution set for the above equation?
Squaring each side o_f x − 12 = √_x + 44 gives (x−12)2 =(√x+44)2 =x+44 x2 +24x+144=x+44 x2 −25x+100=0 (x − 5)(x − 20) = 0 The solutions to the quadratic are x = 5 and x = 20. However, since the first step was to square each side of the given equation, which is not a revers ible operation, you need to check x = 5 and x = 20 in the original equation. Substituting 5 for x gives _ 5 − 12 = √5 + 44 _ −7 = √49
Example 18: A gas in a container will escape through holes of microscopic size, as long as the holes are larger than the gas molecules. This process is called effusion. If a gas of molar mass M1 effuses at a rate of r1 and a gas of molar mass M2 effuses at a rate of r2, then the following relationship holds. This is known as Graham's law. Which of the following correctly.
Suare each side and multiply to simplify. What is left is answer A.
Example 5: What are the solutions x of x^2− 3 = x?
Substituting these formulas into the quadratic formula gives 2 −b±√b −4ac 2 .For the equation x −x−3=0,youhavea=1,b=−1, 2a___ x = −(−1)±√(−1) −4(1)(−3) = 1±√1−(−12) = 1±√13, which is choice D.
Function Notation
The SAT Math Test assesses your understanding of function notation. You must be able to evaluate a function given the rule that defines it, and if the function describes a context, you may need to interpret the value of the function in the context. A question may ask you to interpret a function when an expression, such as 2x or x + 1, is used as the argument instead of the variable x, or a question may ask you to evaluate the composition of two functions.
Example 17: If an object of mass m is moving at speed v, the object's kinetic energy KE is given by the equation KE = 1/2 mv2. If the mass of the object is halved and its speed is doubled, how does the kinetic energy change?
The kinetic energy doubles.
Example 6: If x > 0 and 2x2 + 3x − 2 = 0, what is the value of x?
The left-hand side of the equation can be factored :2x2 +3x−2=(2x−1)(x+2)=0. 1 Therefore,either2x−1=0,which gives x=_,orx+2=0,which givesx=−2. _1 2
Example 13 3x + y = −3 (x + 1)^2− 4(x + 1) − 6 = y If (x, y) is a solution of the system of equations above and y > 0, what is the value of y?
The solutions to the system are (2, −9) and (−3, 6). Since the question states that y > 0, the value of y is 6.
Example 9: Which of the following is equivalent to (1/square root of x)^2
The square root √x sequal to x2.Thus, =x2, and (square root x)=(x2 =x2. Choice B is the correct answer.
Example 14: The graph of which of the following functions in the xy-plane has x-intercepts at −4 and 5?
The x-intercepts of the graph of a function correspond to the zeros of the function. If a function has x-intercepts at −4 and 5, then the values of the function at −4 and 5 are each 0. The function in choice A is in factored form as well.
Example 1: (x^2+ bx − 2)(x + 3) = x^3+ 6x^2 + 7x − 6 In the equation above, b is a constant. If the equation is true for all values of x, what is the value of b?
To find the value of b, expand the left-hand side of the equation and then collect like terms so that the left-hand side is in the same form as the right-hand side. Since the two polynomials are equal for all values of x, the coefficient of matchin powers of x should be the same.Therefore, x^3+(3+b)^x2+(3b−2)x−6 and x^3+6x^2+7x−6 reveals that t3+b=6 and −2=7.Solving either of these equations gives b = 3, which is choice B.