Algebra handout
Rules of exponents:
bn x bm = _________
bn / bm =
_________
(bn)m=
_________
Negative Exponents
With negative exponents, invert the base and make the exponent positive.
Solve the for:
(-6)-3 =
! ! ! !! ! !!
! !! ! ! ! !!
=
Fractional Exponents and Radicals
Radicals can be expressed as exponents, so let’s take
x
n
k
.
v X is the base
v N is the exponent the term is raised to
v K is the root you took of the term
Quick Tips:
If the number under the radical is positive, whether the root is odd or even, there
will always be an answer
If the number under the radical is negative and the root is even, there will not be a
real answer.
If the number under the radical is negative and the root is odd, there will be a real
answer.
Change the following radicals to exponent form:
5
(− 5)7 = ____________
3
(− 7 )4 = ____________
!
4! =
If
!!!!!
!
____________
= 5𝑥 ! , what is the value of x ?
EQUATIONS, EQUATIONS, AND MORE EQUATIONS:
Absolute Value Functions and Inequalities
The absolute value of a term is its distance from zero on the number line.
Solve the following equations:
6x − 40 = 2
If w is an integer and− 𝑤 ! = 𝑤 ! , then which of the following is a possible value of w?
(A) 0
(B) -1
(C) 1
(D) 3
(E) 2
The absolute value functions can also come in the form of inequalities.
Solve the following inequalities:
3x − 9 <6
7x − 21 ≥ 14
Radical equations
A radical equation is an equation in which the variable appears underneath a radical sign.
Solve the following radical equations:
3
2x − 1 − 7 = 2
5 𝑥 − 1 + 13 = 48
Exponential Equations
These equations have variables in the exponents in them.
42x-1=64
!
If 3x = 81 and 2x+y = 64, then ! =
(A) 1
(B) 2/3
(C) 2
(D) 5/2
(E) 3
WELCOME TO FUNCTION LAND!
So far, you’ve learned about exponents, roots, and certain types of equations. Now,
you’re off to learn about functions.
Functions and How to Prove a Graph is a Function
In a function, each x value has a corresponding y value. Because the y value depends on
the x value, y is a function of x.
In order for something to be a function, each x value has only one y value.
On the SAT, a function question can be solved by plugging in a certain term into the
equation. The term can be a number, a variable, or even another function.
If f(x)=x2+2x, find the following
(A) f(3)+f(-3)
(B) f(x+2)
If f(x) = 2x2-3x+1 and g(x) = 3x, find the following
(A) f(g(x))=
(B) f(g(-2))=
(C) g(f(-2))=
Domain and Range:
The domain of a function is all the possible __________________________ of the
function given. The range of the function is all possible _______________________ of
the function given.
Which of the following number is NOT in the domain of f (x)= 4 − 𝑥?
(A) -6
(B) -4
(C) 0
(D) 4
(E) 6
Which of the following is NOT in the range of f (x)=x2-3?
(A) 6
(B) 1
(C) 0
(D) -1
(E) -6
THE COORDINATE PLANE:
A Coordinate Point describes the position of a single point on the graph: ( ____ , ____ )
A Quadrant describes a specific region in the Coordinate Plane; each quadrant is labeled
with a roman numeral.
Quadrant I : x: _____ | y: _____
Quadrant III: x: _____ | y: _____
Quadrant II: x: _____ | y: _____
Quadrant IV: x: _____ | y: _____
Linear Functions
The Slope of a line: y= mx + b
M is the
____________
B is the
____________
The slope of a line can be calculated by plugging in two point (x1,y1) (x2,y2) in the
!!!!!
following formula !!!!!
What is the equation of the line that passes through (3,10) and (2,5)?
(A) y=-5x-5
(B) y=x+5
(C) y=5x-5
(D) y=5x+10
(E) y=5x-10
Label each slope as positive, negative, zero, or undefined:
Parallel Lines: Slopes are: _____________
Perpendicular Lines: Slopes are: _____________________ and
_______________________ .
What is the slope of the line that passes through (3,2) and is parallel to the line that passes
through (-2,3) and (2,-3)?
(A)-3/2
(B) -2/3
(C) 2/3
(D) 1
(E) 3/2
Distance Formula: _____________________________
A circle whose center is at (6,8) passes through (1,1) what is the radius of the circle?
A. 6
B. 8
C. 74
D. 74
E. 24
Transformations
Vertical and horizontal shifts:
v
v
v
v
f(x)+y moves the graph up y units
f(x)-y moves the graph down y units
f(x+y) moves the graph left y units
f(x-y) moves the graph right y units\
The graph of y=f(x) is shown above. Which of the following is the graph of y= f(x-1) –
2?
(A)
(C)
(B)
(D)
(E)