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CLEP Precalculus QUESTIONS AND CORRECT PASSED ANSWERS 2025/2026
Typology: Exams
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Natural Numbers
1, 2, 3, 4, and so on.
exponential growth model
A(t)=A(o)e^rt
whole numbers
0, 1, 2, 3, and so on.
integers
0, ±1, ±2, ±3, and so on.
perfect squares
1 4 9 16 25 36 49 64, and so on.
rational number
A rational number is simply a number of arithmetic: Any whole number, fraction, mixed number, or decimal; together with its negative image.
A rational number can always be written in what form?
As a fraction, where a and b are integers (b not equal to 0).
Which natural numbers have rational square roots?
Only the square roots of the square numbers; that is, the square roots of the perfect squares.
real number
A real number is distinguished from an imaginary or complex number. It is what we call any rational or irrational number.
They are the numbers we expect to find on the number line.
They are the numbers we need for measuring.
function
A rule that relates two variables, typically x and y
domain
The values that x may assume
Let f(x) = x5 and g(x) = x1/
f(g(x))
x
Let f(x) = x5 and g(x) = x1/
g(f(x))
x
parabola
y = x
square root function
y = Square root of x
cubic function
y = x
reciprocal function
y = 1/x
x-intercept
solution
y-intercept
constant
slope intercept form
y = ax + b
quadratic equation
ax2 + bx + c = 0
three methods for solving a quadratic equation
f(x) = x2 −2x −
completing the square
Transpose the constant term to the right
Add a square number to both sides. Add the square of half the coefficient of x.
The left-hand side is now the perfect square
The quadratic formula
Dividend/Divisor
= Quotient + Remainder/Divisor
Dividend
= Quotient· Divisor + Remainder.
The remainder theorem
The value of a polynomial P(x) at x =a,
P(a), is equal to the remainder upon dividing P(x) by x − a
Use synthetic division to divide
x3 − 8x2 + x + 2 by x − 7.
(x2 − x − 6)(x − 7) − 40
The Factor Theorem
x − r is a factor of a polynomial P(x) if and only if r is a root of P(x)
Fundamental Theorem of Algebra
A polynomial of degree n has at least one root, real or complex.
The integer root theorem
If an integer is a root of a polynomial whose coefficients are integers and whose leading coefficient is ±1, then that integer is a factor of the constant term.
Conjugate pairs
If the irrational number a + Square root of b is a root, then its conjugate a − Square root of b is also a root.
Proof of the factor theorem
x − r is a factor of a polynomial P(x)
if and only if
r is a root of P(x).
rigid movement, vertically or horizontally.
absolute value function
y = |x|
If the graph of y= f(x) is translated a units horizontally and b units vertically, then the equation of the translated graph is
y − b = f (x − a)
Write the equation of the parabola whose vertex is at (1, 2)
y − 2 = (x − 1)
Write the equation of the parabola whose vertex is at (−1, 2)
y − 2 = (x + 1)
What are the coördinates of the vertex of this parabola?
y = x2 + 5
The vertex is at (0, 5)
equation of a circle
x2 + y2 = r
Write the equation of the circle of radius 3, and center at the following point. (1, 2)
(x - 1)2 + (y - 2)2 = 9
Write the equation of the circle of radius 3, and center at the following point. (-1, - 2)
(x + 1)2 + (y + 2)2 = 9
If the graph of y = f (x) is translated a units horizontally and b units vertically, then the equation of the translated graph is
y − b = f(x − a)
If we multiply f(x) by a number greater than 1 -- as in the graph in the center
then every y-value is stretched
But if we multiply f(x) by a number less than 1 -- as in the graph on the right
then every y-value is shrunk
A rational function is a quotient of
inverse
reverses the action of that function.
identity function
I(x) = x
logbx = n
bn = x
Write in exponential form: log(2)32 = 5
2^5 = 32
log(b)x + log(b)y
log(b)xy
log(b)x − log(b)y
log(b)x/y
n log(b)x
log(b)x^n
log(b)y = x
b^x = y
f(x)=AsinB(x-c) = 0
|A| = Amplitude of equation
C= Horizontal or phase shift
D= vertical shift
2(pie)/B= fundamental period
square pyramid surface surface area
area of base + 4(area of face)
tangent
= sin/cos
sin
cot
adj/opposite
csc
hypotenuse/opposite
sin-1 (inverse sine function)
Opposite / Hypotenuse
cos- 1
adj/hypotenuse
tan- 1
(Opposite / Adjacent) = θ
g(x)=−b∣x+h∣+g
b=shift
h=(-)horizontal shift
g=vertical shift
