Pre-Calc Test 2 Info

Cubic Function

f(x)=x³

Rational Function

f(x)=1/x

Quadratic Function

f(x)=x²

Exponential Growth Function

f(x)=2^x

Linear Function

f(x)=x

Logarithmic Function

f(x)=log(x)

Absolute Value Function

f(x)=|x|

Square Root Function

f(x)=squrt x

Exponential Decay Function

f(x)=1/2^x

Key Features of Logarithmic Functions

Will always have vertical asymptotes that affect the domain of the function.

Key Features of Rational Functions

Often have vertical and horizontal asymptotes that will affect the domain & range of the function.

Key Features of Exponential Functions

Will always have horizontal asymptotes that affect the range of the function.

Average Rate of Change (ARC

y₂-y₁ / x₂ -x₁ = f(x₂)-f(x₁)/x₂-x₁

Difference Quotient

f(x+h)-f(x) / h. h can’t equal 0

Horizontal Shifts

f(x+h)

f(x-h)

+h=inside moves left

-h=insidemoves right

Vertical Shift

f(x)+k

f(x)-k

+k= outside moves up

-k=outside moves down

Reflects

f(-x)

-f(x)

Multiplying inputs by -1 will reflect over the Y-AXIS

Multiplying outputs by -1 will reflect over the X-AXIS

Vertical Stretch/Shrink

c*f(x)

Multiplying outputs by c >1 is a VERTICAL STRETCH BY A FACTOR OF C

Multiplying outputs by 0<c<1 is a HORIZONTAL STRETCH BY A FACTOR OF 1/C

Horizontal Stretch/Shrink

f(c*x)

Multiplying inputs by c>1 is a HORIZONTAL SHRINK BY A FACTOR OF 1/C

Multiplying inputs by 0<c<1 is a HORIZONTAL STRETCH BY A FACTOR OF 1/C

Function Opperations: ADDITION

(f+g)(x)= f(x)+g(x)

Function Opperations: SUBTRACTION

(f-g)(x)= f(x)-g(x)

Function Opperations: MULTIPLICATION

(fg)(x)=f(x)*g(x)

Function Opperations: DIVISON

(f/g)(x)= f(x)/g(x)

Function Composition

(f o g)(x)= f(g(x))

Vertex Form

(h,k)

f(x)=a(x-h)²+k

Standard Form of Quadratic Functions

y=ax²+bx+c

ax²=quadratic term

bx=linear term

c=constant term

Finding vertex from standard form

1. find x-coordinate with h=-b/2a

2. Plug h value found back into the function to find the K

3. “a” value in vertex form is the same as the “a” value in standard form

Quadratic Formula

x= b+- squrt b²-4ac /2a

Discriminant

b²-4ac

IF:

b²-4ac >0, then there are 2 real x intercepts

b²-4ac = 0, then there is 1 real x intercept

b²-4ac<0, there are NO real x intercepts

Parabola Opening

a<o Parabola opens downward

a>0 Parabola opens upward