Library of Parent Functions

Linear

f(x)=x

Key points: (-1, -1)(0,0)(1,1)

domain: ( -∞, ∞)

Range: ( -∞, ∞)

STF: g(x)=a[b(x+c)]+d

Absolute Value

f(x)= |x|

Key points: (-1,1)(0,0)(1,1)

domain: (-∞, ∞)

Range: [0, ∞)

STF: g(x)=a|b(x+c)|+d

Quadratic

f(x)=x²

Key points: (-2,4)(-2,1)(0,0)(1,1)(2,4)

domain: (-∞, ∞)

Range: [0, ∞)

STF: g(x)=a(b(x+c))²+d

Cubic

f(x)= x³

Key points: (-2,-8)(-1,-1)(0,0)(1,1)(2,4)

domain: (-∞, ∞)

Range: [0, ∞)

STF:g(x)=a(b(x+c))³+d

square root

f(x)=√x

Key points: (0,0)(1,1)(4,2)

domain: (0, ∞)

Range: (0, ∞)

STF:a√b(x+c)) +d

Cube root

f(x)=∛x

Key points: (-8,-2)(-1,-1)(0,0)(1,1)(8,2)

domain: (-∞, ∞)

Range: (-∞, ∞)

STF: g(x)=a∛b(x+c)) +d

reciprocal

f(x)= 1/x

VA: x=0

HA: y=0

Key points: (-1,-1)(0,0)(1,1)

domain: (-∞, 0)u(0, ∞)

Range: (-∞, 0)u(0, ∞)

STF: g(x)=a/(b(x+c)) +d

Rational

f(x)= 1/x²

VA: x=0

HA: y=0

Key points: (-1,1)(0,0)(1,1)

domain: (-∞, 0)u(0, ∞)

Range: (-∞, 0)u(0, ∞)

STF: g(x)=a/(b(x+c))² +d

exponential

f(x)= B^x, with base B

HA: y=0

Key points: (-1, 1/B)(0,1)(1,B)

domain: (-∞, ∞)

Range: (-∞, 0)u(0, ∞)

STF: g(x)=a(B)^(b(x+c)) +d

logarithmic

f(x)=LogBx

VA: x=0

Key points: (1/B)(1,0)(B,1)

domain: (0, ∞)

Range: (-∞, ∞)

STF: g(x)=alogB[b(x+c)]+d

Natural Base Exponential

f(x)=e^x

HA: y=0

Key points: (-1, 1/e)(0,1)*(1,e)

domain: (-∞, ∞)

Range: (-∞, 0)u(0, ∞)

STF: g(x)=a(e)^(b(x+c)) +d

Natural Logarithmic

f(x)=lnx

VA: x=0

Key points: (1/e, -1)(1,0)*(e,1)

domain: (0, ∞)

Range: (-∞, 0)u(0, ∞)

STF: g(x)=a ln [b(x+c)]+d

Sine

f(x)=sinx

Key points:(0,0)(π/2, 1)(π,0)(3π/2, -1)(2π, 1)

domain: (-∞, ∞)

Range: [-1, 1]

STF: g(x)=a sin [b(x+c)]+d

cosine

f(x)=cosx

Key points: (0,1)(π/2, 0)(π, -1)(3π/2, 0)(2π, 1)

domain: (-∞, ∞)

Range: [-1, 1]

STF: g(x)=a cos [b(x+c)]+d

tangent

f(x)=tanx

VA: x= π/2 + πk, kEZ

key points: (-π/4, -1)(0,0)(π/4, 1)

Domain: {x| x ≠ π/2 +πn}

Range: (-∞, ∞)

STF: g(x)=a tan [b(x+c)]+d

Cosecant

f(x)=cscx

VA: x= πk, kEZ

Key points: (π/2, 1)(3π/2, -1)

domain: (-∞, ∞)

Range: [-1, 1]

STF: g(x)=a csc [b(x+c)]+d

secant

f(x)=secx

VA: x= π/2 +πk, kEZ

Key points: (0,1)(π, -1)(2π, 1)

domain: {x| x ≠ π/2 +πn}

Range: (-1, 1)

STF: g(x)=a sec [b(x+c)]+d

cotangent

f(x)=cotx

VA: x= πk, kEZ

Key points: (π/4, 1)(π/2, 0)(π/4, -1)

domain: (-∞, ∞)

Range: [-1, 1]

STF: g(x)=a cot [b(x+c)]+d