Knowing certain relationships and formulas that are not provided on the reference page of the test booklet can increase your speed and accuracy.
Example :: No-Calculator Section :: Grid-In
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) represents a polynomial and the value of f(c) is r, then r is the remainder when f(x) is divided by x − c. If r = 0, then 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 encountered Pythagorean triples such as 3-4-5, 5-12-13, 8-15-17, and 7-24-25.
You are expected to know the right triangle definitions of the sine, cosine, and tangent 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 area of an overlapping shaded region usually involves subtracting the area of one familiar type of figure from a larger one.