If (p + q)2 = 78 and (p − q)2 = 50, what is the value of pq?
Solution
Square each binomial and then subtract corresponding sides of the two equations:
Grid-in 7
STRATEGY 15: REWRITE EXPRESSIONS OR EQUATIONS IN EQUIVALENT WAYS
Some SAT Math questions may ask you to find the value of an algebraic expression using an accompanying equation. By rewriting the given expression or equation in an equivalent way, you may be able to make a substitution that will produce its value.
Example :: No-Calculator Section :: Multiple-Choice
If 2x − 3y = 10, what is the value of ? (A)
(B) 210 (C) 46 (D) 85 Solution
■ Write the fraction in an equivalent way by expressing the numerator and denominator as a power of the same base:
■ Since it is given that 2x − 3y = 10, substitute 10 for the exponent:
The correct choice is (B).
Example :: No-Calculator Section :: Grid-In
If , what is the value of 3a + 4b?
Solution
Write the equation in an equivalent way by eliminating its fractional coefficients. Multiply each term on both sides of the equation by 6, the lowest common multiple of 2 and 3:
Grid-in 30
Example :: No-Calculator Section :: Grid-In
A population of rabbits doubles every 48 days according to the formula , where P is the population of rabbits on day t. What is the value of t when the population of rabbits is 320?
Solution
If P = 320, then Since 25 = 32, which means = 5 so t = 48 × 5 = 240.
Grid-in 240
STRATEGY 16: KNOW HOW TO DO THE MATH
Some SAT test questions simply require a straightforward application of some routine algebraic technique, geometric fact, or trigonometric relationship that you studied in your regular mathematics classes.
Example :: No-Calculator Section :: Multiple-Choice
If p and q satisfy 2y2 + y = 21 and p > q, which of the following is the value of p − q?
TIP
You are re quire d to know how to factor quadratic trinomials of the form ax2 + bx + c whe re the constant a, the coe fficie nt of the x2-te rm, is diffe re nt than 1. You should also know how to solve nonfactorable quadratic e quations e ithe r by
comple ting the square or by using the quadratic formula.
Solution
Rewrite the given quadratic equation as 2y2 + y − 21 = 0 and solve it by factoring using the reverse of FOIL:
Since p = 3 and q = – ,
The correct choice is (B).
Example :: No-Calculator Section :: Multiple-Choice
An acute angle of a right triangle measures x radians. Which of the following is equal to cos x when x =
Solution
Two angles are complementary if their measures add up to 90° The cosine of an acute angle is equal to the sine of the complement of that angle. In general, for acute angles,
Substitute radians for x:
The correct choice is (D).
Example :: No-Calculator Section :: Grid-In
In the equation above, what is the value of y?
Solution
■ Change the radicals to “i-form”:
■ Multiply the binomials horizontally using FOIL:
■ Since 6 + 7i = x + yi, y = 7.
Grid-in 7
Example :: No-Calculator Section :: Grid-In
A quadratic function and a linear function are graphed in the xy-plane as shown above. The vertex of the graph of the quadratic function is at (3, 25). If the two graphs intersect in the first quadrant at the point (j, k), what is the value of the product jk?
Solution
■ The graph of the quadratic function is a parabola. If the vertex of a parabola is at (h, k), then the equation of the parabola has the form y = a(x −h)2 + k. Since it is given that (h, k) = (3, 25), y = a(x
−3)2 + 25. From the graph, you know that the y-intercept is (0, 16). Plug these coordinates into the equation to find the value of the constant a:
Hence, the equation of the parabola is y = −(x − 3)2 + 25.
■ Find an equation of the line in y = mx + b slope-intercept form where m is the slope of the line and b is its y-intercept. The y-intercept of the line is −5 so b = −5. Using the points (4, 3) and (0, −5), find the slope, m, of the line:
Hence, an equation of the line is y = 2x − 5.
■ To find the coordinates of the point of intersection of the two graphs, solve the linear-quadratic system of equations:
Substitute 2x − 5 for y in the quadratic equation:
Since (j, k) is in the first quadrant, j = 7. Find the corresponding value of y by substituting 7 for x in the linear equation, which gives y = 2(7) − 5 = 9. Hence, k = 9 so jk = 7 × 9 = 63.
Grid-in 63
STRATEGY 17: KNOW HOW TO WORK WITH FUNCTIONS
Functions and function notation are commonly used on the SAT. A function is simply a mathematical rule that describes how to use an input value, say x, to produce an output value, say y. A function may take the form of either a “y = . . .” type of equation, a graph, or a table. If a function named f is described by the equation y = x2 − 4, then in function notation, f(x) = x2 − 4. The notation f(3) represents the “y-value” of function f when x = 3 so
The ordered pair (3, 5) belongs to function f and is a point on its graph.
Example :: No-Calculator Section :: Grid-In
If function h is defined by h(x) = ax2 − 7 and h(−3) = 29, what is (A) −6
(B) −5 (C) 4 (D) 8 Solution
■ Use the fact that h(−3) = 29 to find the value of the constant a:
h(−3) = a(−3)2 − 7 = 9a − 7 = 29 so 9a = 36 and a = 4
■ Evaluate
The correct choice is (A).
Example :: No-Calculator Section :: Multiple-Choice
If function f is defined by f(x) = 3x − 4, then f(−2x) = (A) −6x + 4
(B) 8 − 6x (C) −6x − 4 (D) 8x + 6x2 Solution
The correct answer choice is (C).
Example :: No-Calculator Section :: Grid-In
The graph of function f over the interval −5 ≤ x ≤ 5 is shown in the figure above. If f(w) = 2 and w > 0, what is one possible value of w?
Solution
The notation f(w) = 2 represents the point (w, 2) on the graph of function f. Each point whose x-coordinate is between 0 and 1 has a corresponding y-coordinate of 2. Hence, w can be any number between 0 and 1.
For example, if Grid-in 1/3
Example :: No-Calculator Section :: Multiple-Choice
Based on the graph in the previous example, which of the following statements must be true?
I. f(5) + f(−5) = 0.
II. If −5 ≤ x ≤ 5, the maximum value of function f is 4.
III. The equation f(x) = 3 has 3 real solutions.
(A) I only (B) II only (C) I and II only (D) II and III only Solution
■ f(−5) = 0 and f(5) = 1 so f(5) + f(−5) ≠ 0. Statement I is false.
■ The highest point on the graph is (−3, 4) so the maximum value of function f is 4. Statement II is true.
✓
■ The horizontal line y = 3 intersects the graph at 2 points so f(x) = 3 has 2 rather than 3 real solutions.
Statement III is false.
Since only Statement II is true, the correct choice is (B).
Example :: No-Calculator Section :: Multiple-Choice
The table above shows a few values of the linear function f. Which of the following equations defines f ? (A) f(x) = 2x − 3
(B) f(x) = −2x + 3 (C) f(x) = −3x + 2 (D) f(x) = 3x − 2 Solution
The graph of a linear function is a line. From the table, each time x increases by 2 units, y decreases by 6 units so the slope of the line is = –3. Hence, the coefficient of x for
a linear function of the form f(x) = mx + b is −3. Only the function in choice (C) satisfies this condition.
The correct choice is (C).
Example :: No-Calculator Section :: Multiple-Choice
Function g is related to function f by the equation g(x) = f(x − 1) − 2. If the point (4, 3) is on the graph of function f, what are the coordinates of the corresponding point on the graph of function g?
(A) (3, 1)
■ Evaluate the “inside” function first: h(3) = 8 − 32 = 8 − 9 = −1.
■ Then
A parabola y = ax2 + bx + c with a > 0 passes through the points (−2, 3), (p, q), and (5, 3). If (p, q) is the lowest point on the parabola, what is the value of p?
Solution
Draw a diagram:
The line of symmetry is the perpendicular bisector of any horizontal segment whose endpoints are on the parabola. Since the points (−2, 3) and (5, 3) are corresponding parabola points on either side of the axis of symmetry, the line of symmetry bisects the segment joining these points and also passes through the vertex of the parabola. Hence, the x-coordinate of (p, q) must be the same as the x-coordinate of the midpoint of this segment:
Grid-in 3/2
Example :: Calculator Section :: Multiple-Choice
Note: Figure not drawn to scale.
The graph of f(x) = −0.5x2 + x + 4 in the xy-plane is the parabola shown in the figure above. The parabola crosses the x-axis at D(−2, 0) and at point C(x, 0). Point A is the vertex of the parabola. Segment AC and the line of symmetry, AB, are drawn. What is the number of square units in the area of ABC?
(A) 4.5 (B) 6.25 (C) 6.75 (D) 13.5 Solution
■ A math fact that you should know is that the x-coordinate of the vertex of the parabola y = ax2 + bx + c can be determined using the formula, x = –
To find the y-coordinate of the vertex, substitute 1 for x in the parabola equation:
Hence, AB = 4.5.
■ The coordinates of point B are (1, 0) so DB = 1 − (−2) = 3. Since the line of symmetry is the perpendicular bisector of segment CD, DB = BC = 3.
■ The area of a right triangle is one-half of the product of the lengths of its legs:
The correct choice is (C).
Example :: No-Calculator Section :: Multiple-Choice
I. f(x) is divisible by x − 5.
II. In the interval 0 ≤ x ≤ 6, exactly one x–value satisfies the equation f(x) = 5.
III. (x + 2) is a factor of f(x).
The diagram above shows the graph of a polynomial function f. Which statement or statements in the box above must be true?
(A) II only (B) III only (C) I and II only (D) II and III only Solution
The SAT assumes you know some basic facts related to polynomial functions and their graphs. If the graph of a polynomial function f(x) intersects the x-axis at a point whose x-coordinate is c, then the following statements are equivalent and interchangeable:
■ f(c) = 0.
■ c satisfies the equation f(x) = 0.
■ x − c is a factor of f(x).
■ f(x) is divisible by x − c.
TIP
If f(x) re pre se nts a polynomial and the value of f(c) is r, the n r is the re mainde r whe n f(x) is divide d by x − c. If r = 0, the n f(x) is divisible by x − c.
Consider each of the answer choices in turn:
■ Since the graph does not intersect the x-axis at x = 5, f(x) is not divisible by x − 5. Statement I is false.
■ In the interval 0 ≤ x ≤ 6, the line y = 5 (not drawn) intersects the graph at only one point so there is exactly one value of x that satisfies the equation f(x) = 5. Statement II is true. ✓
■ Since the graph crosses the x-axis at x = −2, x − (−2) or x + 2 is a factor of f(x). Statement III is true.
✓
Since only Statements II and III are true, the correct choice is (D).
Example :: No-Calculator Section :: Multiple-Choice
p(x) = 4(−x3 + 11x + 12) − 6(x − c)
In the polynomial function p(x) defined above, c is a constant. If p(x) is divisible by x, what is the value of c?
(A) −8
(B) −6 (C) 0 (D) 6 Solution
A polynomial function p(x) has the general form
p(x) = anxn + an−1xn + an−3xn−3 + . . . + k
If p(x) is divisible by x, then the value of the constant term, k, must be 0. Otherwise, there would be a remainder.
■ Write p(x) in standard form by removing the parentheses and collecting like terms:
■ Set the constant term equal to 0:
The correct choice is (A).
Example :: Calculator Section :: Grid-In
In the figure above, what is the value of cos x − sin x?
TIP
You should know commonly e ncounte re d Pythagore an triple s such as 3-4-5, 5-12-13, 8-15-17, and 7-24-25. You are e xpe cte d to know the right triangle de finitions of the sine , cosine , and tange nt functions.
Solution
Since the measures of the base angles are equal, the triangle is isosceles so dropping a perpendicular from the vertex to the opposite side bisects the base and creates two smaller right triangles with side lengths that form a 8-15-17 Pythagorean triple:
Hence,
Grid-in 7/17
Example :: No-Calculator Section :: Grid-In
In the xy-plane above, O is the center of the circle, and the measure of ∠AOB is kπ radians. What is a possible value for k ?
Solution
so AOC is a 30º-60º right triangle with m∠AOC = 60.
Hence, ∠AOB measures 120 degrees or, equivalently, π radians.
Grid-in 2/3 TIP
Finding the are a of an ove rlapping shade d re gion usually involve s subtracting the are a of one familiar type of figure from a large r one .
STRATEGY 19: KNOW HOW TO FIND AREAS INDIRECTLY
In order to find the area of a particular region, it may be necessary to find it indirectly by finding the difference between the areas of figures that overlap.
Example :: No-Calculator Section :: Multiple-Choice
The figure above shows a quarter of a circle with center at O. If the length of diagonal DM of rectangle ODCM is 8 and M is the midpoint of OB, what is the area of the shaded region?
Solution
The area of the shaded region is the difference between the areas of the quarter circle and right triangle DOM.
■ Since the diagonals of a rectangle have the same length, OC = DM = 8:
■ OC is a radius of the quarter circle. Hence, the area of the quarter circle is .
■ Since segment OB is also a radius, OB = 8. It is given that M is the midpoint of OB so OM = MB = 4. In right DOM, use the Pythagorean theorem to find the length of OD:
■ The area of right Hence,
The correct choice is (C).
Example :: No-Calculator Section :: Grid-In
What is the number of square units in the area of the quadrilateral shown in the figure above?
Solution
■ Draw line segments to form right triangles at the upper and lower corners of the quadrilateral:
■ Find the area of the quadrilateral indirectly by subtracting the sum of the areas of the two right triangles from the area of square ABCD:
Grid-in 17
Example :: Calculator Section :: Multiple-Choice
Note: Figure not drawn to scale.
The figure above shows a logo in the shape of overlapping equilateral triangles ABC and DEF. If AD = DC = CF = 4, what is the area of the shaded region?
(A) 24 – 8 (B) 28 (C) 32
(D) 36 – 8 Solution
■ AC = DF = 4 + 4 = 8
■ Since, m∠EDF = m∠BCA = m∠DGC = 60, DGC is equilateral with DC = 4.
■ The area of the shaded region is the difference between the sum of the areas of the two overlapping equilateral triangles and equilateral DGC. To find the area of the overlapping equilateral triangles, use the formula
The correct choice is (B).