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Function Families

Assume a, k, and h are positive numbers.

Parent	y = f ( x ) y equals f open x close
Reflection across x-axis	y = − f ( x ) y equals negative f open x close

Vertical stretch ( a > 1 ) open eh greater than 1 close	y = a f ( x ) y equals eh f open x close
Vertical shrink ( 0 < a < 1 ) open 0 less than eh less than 1 close

Translation

horizontal to left by h	y = f ( x + h ) y equals f open x plus h close
horizontal to right by h	y = f ( x − h ) y equals f open x minus h close
vertical up by k	y = f ( x + k ) y equals f open x plus k close
vertical down by k	y = f ( x − k ) y equals f open x minus k close

Chapter 4 Quadratic Functions and Equations

Quadratic Functions

Parent	y = x 2 y equals , x squared
Reflection across x-axis	y = − x 2 y equals negative , x squared

Stretch ( a > 1 ) open eh greater than 1 close	y = a x 2 y equals eh , x squared
Shrink ( 0 < a < 1 ) open 0 less than eh less than 1 close

Translation

horizontal by h vertical by k

y = ( x − h ) 2 + k y equals . open x minus h close squared . plus k

Vertex Form	y = a ( x − h ) 2 + k y equals eh . open x minus h close squared . plus k
Standard Form	y = f ( x ) = a x 2 + b x + c y equals f . open x close equals eh x squared . plus b x plus c

The graph is a parabola that opens up if a > 0 eh greater than 0 and down if a < 0 . eh less than 0 .

The vertex is (h, k) (Vertex Form) and ( − b 2 a , f ( − b 2 a ) ) ( open . negative , fraction b , over 2 eh end fraction , comma f . open . negative , fraction b , over 2 eh end fraction . close . close . open Standard Form).

The axis of symmetry is x = h x equals h (Vertex Form) and x = − b 2 a ( x equals negative , fraction b , over 2 eh end fraction , open Standard Form).

Factoring Perfect-Square Trinomials

a 2 + 2 a b + b 2 = ( a + b ) 2 eh squared , plus 2 eh b plus , b squared , equals . open eh plus b close squared

a 2 − 2 a b + b 2 = ( a − b ) 2 eh squared , minus 2 eh b plus , b squared , equals . open eh minus b close squared

Factoring a Difference of Two Squares

a 2 − b 2 = ( a + b ) ( a − b ) eh squared , minus , b squared , equals open eh plus b close open eh minus b close

Multiplication Property of Square Roots

For any numbers a ≥ 0 eh greater than or equal to 0 and b ≥ 0 , a b = a ⋅ b . b greater than or equal to 0 comma . square root of eh b end root , equals square root of eh dot square root of b .

Division Property of Square Roots

For any numbers a ≥ 0 eh greater than or equal to 0 and b > 0 , a b = a b . b greater than 0 comma . square root of eh over b end root , equals , square root of eh over b end root , .

Zero-Product Property

If a b = 0 , eh b equals 0 comma then a = 0 eh equals 0 or b = b equals 0.

The Quadratic Formula

If a x 2 + b x + c = 0 , eh , x squared , plus b x plus c equals 0 comma then x = − b ± b 2 − 4 a c 2 a x equals . fraction negative b plus minus . square root of b squared , minus 4 eh c end root , over 2 eh end fraction

Discriminant

The discriminant of a quadratic equation in the form a x 2 + b x + c = 0 eh , x squared , plus b x plus c equals 0 is b 2 − 4 a c . b squared , minus 4 eh c .

b 2 − 4 a c > 0 ⇒ b squared , minus 4 eh c greater than 0 rightwards double arrow two real solutions

b 2 − 4 a c = 0 b squared , minus 4 eh c equals 0 ⇒ one real solution

b 2 − 4 a c < b squared , minus 4 eh c less than 0 ⇒ two complex solutions

Square Root of a Negative Real Number

For any positive number a, − a = − 1 ⋅ a = − 1 ⋅ a = i a . square root of negative eh end root , equals . square root of negative 1 dot eh end root . equals , square root of negative 1 end root , dot square root of eh equals i square root of eh .

Example: − 5 = i 5 square root of negative 5 end root , equals i square root of 5

Note that ( − 5 ) 2 = ( i 5 ) 2 = i 2 ( 5 ) 2 = − 1 ⋅ 5 = − 5   ( not  5 ) . open , square root of negative 5 end root , close squared . equals . open , i square root of 5 , close squared . equals , i squared . open square root of 5 close squared . equals negative 1 dot 5 equals negative 5 . open , not , 5 , close . .

Chapter 5 Polynomials and Polynomial Functions

End Behavior of a Polynomial Function

The end behavior of a polynomial function of degree n with leading term a x n : eh , x to the n , colon

a	n	end behavior
positive	even	up and up
positive	odd	down and up
negative	even	down and down
negative	odd	up and down

Factor Theorem

The expression x − a x minus eh is a linear factor of a polynomial if and only if the value a is a zero of the related polynomial function.

Remainder Theorem

If you divide a polynomial P(x) of degree n ≥ 1 n greater than or equal to 1 by x − a , x minus eh comma then the remainder is P(a).

Factoring a Sum or Difference of Cubes

a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 ) eh cubed , plus , b cubed , equals open eh plus b close open , eh squared , minus eh b plus , b squared , close

a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 ) eh cubed , minus , b cubed , equals open eh minus b close open , eh squared , plus eh b plus , b squared , close

Rational Root Theorem

Let P ( x ) = a n x n + a n − 1 x n − 1 + p open x close equals , eh sub n , x to the n , plus eh , n minus sub 1 , x , n minus to the first , plus … + a 1 x + a 0 eh sub 1 , x plus , eh sub 0 be a polynomial with integer coefficients.

Integer roots of P(x ) = 0 close equals 0 must be factors of a 0 . eh sub 0 , .

Rational roots have reduced form p q p over q where p is an integer factor of a 0 eh sub 0 and q is an integer factor of a n . eh sub n , .

Conjugate Root Theorems

Suppose P(x) is a polynomial with rational coefficients.

If a + b eh plus square root of b is an irrational root with a and b rational, then a − b eh minus square root of b is also a root.

Suppose P(x) is a polynomial with real coefficients.

If a + bi is a complex root with a and b real, then a − b i eh minus b i is also a root.

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