Precalculus

Front 1

Graphing inequalities involving a function
1. rewrite the equation as a function f(x) = ...
2. find the x and y intercepts
3. find the axis of symmetry
4. find the vertex
5. determine if the graph opens up or down
6. determine an additional point using the y-intercept and the axis of symmetry

Back 1

2. x-intercept let f(x) = 0
y-intercept let x = 0
3. axis of symmetry -b/2a
4. vertex (-b/2a, f(-b/2a))
5. if a is posative, the graph opens up and the vertex is the minimal point
if a is negative, the graph opens down and the vertex is the maximal point

Front 2

Solving polynomial and rational inequalities
Solve by graphing the function on a number line:
1. rewrite the equation as a function f(x) = ax + bx + c
2. find the x and y intercepts
3. plot the points on a number line using parentheses and brackets that open in both directions
4. list the intervals
5. choose a test number (t) for each interval
6. put the equation into the calculator and use the table to find f(t)
7. determine if the f(t)s are posative or negative
8. circle the intervals that match the equation and shade in the graph

Back 2

3. if the equation is strict (><), use parentheses
if the equation is nonstrict (><), use brackets
if the equation is rational and there is a variable in the denominator of the equation, solve for it and plot the point using parentheses as it will by an asymptote and therefore not included
8. if the equation is > or > then all the intervals where f(x) is posative will be correct answers and vice versa
note: if the equation has separate answers, be sure to use "or" in set-builder notation and "U" between interval notations

Front 3

Solving inequalities involving quadradic functions
1. rewrite the function as f(x) = ...
2. find the x-intercepts
3. plot the x-intercepts on a number line
3. if the inequality is < or < than the the answer is an and problem (a < x < b) a line segment in the center of the graph will be shaded
if the inequality is > or > then the answer is an or problem ( x < a or x > b) two rays going to either infinity will be shaded

Back 3

Methods to solving quadradic functions:
(aka finding the zeros)
1. GCF
x + 4x = x(x + 4)
x = 0 x = -4
2. factoring
x -3x -10 = (x - 5)(x + 2)
x = 5 x = -2
3. square root method
x - 9 = 0
x = 9
x = 3
4. completing the square
x + 4x = 0
x + 4x + 4 = 0 + 4
(x + 4) = 4
x + 4 = 2
x = -2 x = -6
5. quadratic formula
x = -b b - 4ac
2a

Front 4

The descriminant
Interpreting a graph
1. b -4ac > 0 2 x-intercepts
2. b -4ac = 0 vertex is tangent to x-axis at the x-intercept
3. b -4ac < 0 graph does not touch the x-axis

Back 4

The descriminant
Predicting solutions
1. b -4ac > 0, the equation has two unequal real solutions
2. b -4ac = 0, the equation has a repeated real solution, double root
3. b -4ac < 0, the equation has two complex solutions that are not real. The complex solutions are conjugates of each other.

Front 5

quadradic equations
standard form
f(x) = ax + bx + c
general form
f(x) = a(x - h) + k
axis of symmetry
x = h
vertex
(h,k)

Back 5

parabolas - the graphs of all quadratic functions
vertex - the lowest or highest point of a parabola
axis of symmetry - the vertical line passing through the vertex of a parabala
maximum value - f(-b/2a) if a < 0
minimum value - f(-b/2a) if a > 0

Front 6

Absolute value
1. lul = a
u = a or u = -a

2. lul < a
-a < u < a

3. lul > a
u < -a or u > a

Back 6

Things to remember:
-always get the absolute value alone before solving
-when multiplying or dividing by a negative, switch the sign
-answers can be written 3 different ways
1. graph of a number line
2. interval notation
3. set-builder notation

Front 7

Basic transformation charts

x x x lxl
-3 9 -3 3
-2 4 -2 2
-1 1 -1 1
0 0 0 0
1 1 1 1
2 4 2 2
3 9 3 3

Back 7

Basic transformation charts

x x x x
0 0 -2 -8
1 1 -1 -1
4 2 1 1
9 3 2 8

Front 8

Polynomial properties
writing polynomial from factored form
1. solve for each given zero
ex: x = -2 becomes (x + 2)
2. multiply out the quantities and simplify
ex: (x + 2)(x - 2)(x - 3)
(x - 4)(x - 3)
x - 3x - 4x + 12
Note: be careful to watch out for multiples
ex: x + 3 with a multiple of 2 becomes (x - 3)(x - 3)

Back 8

Polynomial properties
3 degree polynomials
interpreting a graph
1. even multiplicity - the graph will be tangent to the x-axis at one point and cross at another
2. odd multiplicity - the graph will cross the x-axis at 3 points
3. odd multiplicity with x = 0 - the graph will go through the origin

Front 9

transformations
f(x) = x

3x stretch
1/2x compress
-x reflects over x-axis
x + 3 shifts up
x - 5 shifts down
(x + 3) shifts left
(x - 5) shifts right

Back 9

f(x) = x

a(x - h) + k

a - stretches, compresses, or reflects
h - shifts right or left(the negative sign reverses the direction)
k - shifts up or down