exam 1

Question 1

Find the slope of two points

Answer 1

(y1-y2) / (x1-x2) = m

Question 2

Find an equation using a point and a slope

Answer 2

y-y1 = m * (x-x1)
OR
y = mx + b

Question 3

Find domain

Answer 3

even roots cannot contain a negative value
AND
Denominators cannot equal 0

Question 4

Find range of rational functions

Answer 4

Graph the equation,
When numerator factors equal zero those are x intercepts
When denominator factors equal zero those are holes or vertical asymptotes. When a domain issue gets canceled out it becomes a hole. Everything that happens outside of the fraction just affects the output.

Question 5

Graphing a function

Answer 5

(x-1) is a horizontal shift right 1
(x) -1 is a vertical shift down 1
2(x) multiply the output by two (expand vertically)
(2x)^2 multipy the input by 2, squeeze horizontal
-(x)^2 flips the function over the x axis
(-x)^2 flips the function over the y axis
Basically, if it happen before the power or square root or fraction, it affects the input, if it happens afterwards, it affects the output

Question 6

Find limits

Answer 6

Keep the limit in front of the function until you evaluate
If function evaluates to 0/0 there is most likely a factor you can cancel out like a hole

Question 7

End behavior

Answer 7

Determined by largest power factors, divide the numerator by the denominator, keep signs

Question 8

Sign test

Answer 8

Enter values approaching the denominator zero from the left and right to determine what infinity the asymptote goes to

Question 9

Difference Quotient

Answer 9

Basically slope, m(v)sec
[f(x+deltax) - f(x)] / deltax

Question 10

Slope of a curve at a point

Answer 10

Difference quotient but with a limit in front of it having deltax approach 0 (ALSO CALLED THE DERIVATIVE OR F PRIME OF X)

Question 11

Derivative f’(x) or d/dx

Answer 11

(Formula found to evaluate the slope of a curve at any point along that curve) Evaluate through the limit definition of a derivative (slope of a curve at a point, m(v)tan, diff quotient with limit ect.)
OR
Through the rules of derivatives (If x is an integer)
f(x) = x^n THEN f’(x) = n*x^n-1

Question 12

x^2 end behavior

Answer 12

upward pointing parab

Question 13

x^3 end behavior

Answer 13

Downward point when x is neg upward point when x is pos

Question 14

sqrroot(x) end behavior

Answer 14

start at origin, 0,0 > 1,1 > 4,2

Question 15

1/x end behavior

Answer 15

vertical asymptote when denom=0 and horizontal asymptote at y=0

Question 16

Sin =
Cos =
Tan =
Csc =
Sec =
Cot =

Answer 16

Sin = y/1
Cos = x/1
Tan = y/x
Csc = 1/y
Sec = 1/x
Cot = x/y

Question 17

Limit rules:

Answer 17

One limit can be seperated into two limits by addition subtract and multiplication
Also can be done through division as long as denominator does not equal 0.
The limit of a constant is simply that constant
The limit of a function rasied to a power is equal to raising the limit of the function to a power: lim [f(x)]^2 = [lim f(x)]^2
Same thing with the power for square roots
The limit of a polynomial as x approaches a is equal to the polynomial evaluated to a (i.e. just plug the limit value in)
You can plug limit values into any function to see if it works, if it evaluates to a real number, that is your limit.
When you evaluate for a limit and it results to 0/0 you know that the number you evaluated is a factor somewhere in that rational function.
Limits must be equal from right and from left to exist, if not they are DNE (does not exist)
If you can cancel out the domain issue, do a sign analysis test.

Question 18

Continuity

Answer 18

Draw the line between two points without having to lift it up, no holes, no asymptotes, no jumps

Question 19

Trig limit rules

Answer 19

lim x->a sin(x) = sin(a)
lim x->a cos(x) = cos(a)
lim x->a tan(x) = tan(a) AS LONG AS cos(a) does not equal 0
Trig limits can be split up through the 4 operators just like regular limits as long as the denominator doesn’t equal 0 with division
the same power rule applies where you can take the power out of the function and just apply it to the whole limit