Pre-Calculus Chap 1

Algebraic Function 1.1

Formed by applying algebraic operations to linear function

ex.

Linear f(x)=x

Quadratic f(x)=x²

Cubic f(x)=x³

Rational f(x)=1/x

Square Root f(x)=√x

Transcendental Function 1.1

Can’t be formed from f(x)=x with algebraic operations

ex.

Exponential f(x)=a^x, a>0, a≠1

Logarithmic f(x)=log_ax, a>0, a≠1, x>0

Trigonometric f(x)=sinx, f(x)=cosx, f(x)=tanx, f(x)=cscx, f(x)=secx, f(x)=cotx

Inverse Trigonometric f(x)=arcsinx, f(x)=arccosx

Nonelementary Function 1.1

A function that can't be formed with just arithmetic, exponents, logs, and trig functions(Neither Algebraic nor Transcendental)

ex.

Absolute Value f(x)=|x|

Greatest Integer f(x)=⌊x⌋

General Form of a Line 1.1

Ax+By+c=0

any line can be written in the form

can be used to write line where the slope is undefined

Vertical Line Equation 1.1

x=a

Horizontal Line 1.1

y=b

Slope-Intercept Form 1.1

y=mx+b

Point Slope Form 1.1

y-y₁ = m(x-x₁)

Even Function 1.3

symmetrical with respect to the y-axis

f(-x)=f(x)

Odd Function 1.3

Symmetrical with respect to the origin

f(-x)=-f(x)

graph that is symmetrical with respect to the x-axis and is still a function 1.3

y=0

Domain of an Arithmetic Combination of Functions f&g 1.5

All real numbers that are common to the domain of both f&g

Domain of the composition of 2 functions (f ∘ g)(x) 1.5

all x in the domain of g such that g(x) is in the domain of f

*composition of (f ∘ g) is generally not equal to (g ∘ f)

Inverse Functions 1.6

A function that reverses(undoes) another function

Notation: f^-1

ex. f(x)= x+4, f^-1(x)=x-4

• f^-1(x) is the reflection of f over the line y=x

• domain of f= range of f^-1

• f(f^-1(x))=x and f^-1(f(x))=x

• a function must be one-to-one to have an inverse

Linear Regression 1.7

used to find the line of best fit for points

• r = correlation coefficient, closer to 1 |r| is, the more accurate the model is

Sum of squared differences 1.7

used to show how close a linear model is to the actual data

least squares regression line 1.7

model with the lowest sum of squared differences

how do you find line of best fit 1.7

on a calculator: use the linear regression feature

by hand: compare the model against the actual points with the sum of squared differences

How do you convert standard form to Vertex Form (rewrite a quadratic as the square of a binomial) 2.1

Completing the Square by…

take standard form

• ax² + bx +c

move the constant term to the other side of the equation

• ax² + bx +c —> ax² + bx = -c

divide everything by a

• ax² + bx = -c —> (ax² + bx)/a = (-c)/a

find the new b term

• (b/2)^2

add the new b term to both sides of the equation

• (ax² + bx)/a = (-c)/a —> x² + (bx)/a + (b/2)^2 = (-c)/a + (b/2)^2

factor the left side by rewriting it as a perfect square trinomial

• x² + (bx)/a + (b/2)^2 = (-c)/a + (b/2)^2 —> (x-#)(x-#) = (-c)/a + (b/2)^2

• #s add to (bx)/a and multiply to (b/2)^2

move the right side back to the left side

• y = (x-#)² - ((-c)/a + (b/2)^2)

• y=(x-h)² + k