[graphing part 1][section 14]
(1.) y = x2 + x - 30 here is the problem
0 = (x + 6)(x - 5) factor and set equal to 0
x + 6 = 0 x - 5 = 0
set each factor equal to 0
-6 -6 +
5 +5 add this to each side
___________ ______________
x = -6 , x
= 5 add
(ii.) y = 02 + 0 - 30
replace x with 0
y = -30 combine like terms
(iii.) y' = 2x + 1 take the derivative
2x + 1 = 0 set the derivative equal to 0
-1 -1
subtract 1 from each side
_________________
2x =
-1 subtract
____ ____
2
2 divide each side by 2
x = -1/2 cancel
y(-1/2) = (-1/2)2 + (-1/2) -
30 replace x with -1/2
y(-1/2) = -30.25 multiply and combine like terms
results: (-6,0) and (5,0) are the x
intercepts
(0, -30) is the y intercept
(-1/2, -30.25) is the relative
minimum
(2.) y = -x2 + 2x - 4
(i.) y =
-(0)2 + 2(0) - 4 replace x
with 0
y = -4 multiply combine like terms
(ii.) y' = -2x + 2 take the derivative
-2x + 2 = 0
set the derivative equal to 0
2x - 2 = 0 multiply thru by -1
+ 2
+2 add 2 to each side
_______________
2x =
2 add
____ ____
2 2
divide each side by 2
x = 1 divide and cancel
y(1) = -(1)2 + 2(1) - 4
replace x with 1
y(1) = -3 multiply combine like
terms
results: (0, -4) is the y intercept
(1,-3) is the relative maximum point
(3.) y= 4x2 - 8 here is the problem
y = 4(0)2 - 8 replace x with 0
y = -8 multiply combine like terms
(ii.)
4x2 - 8 = 0 set the
function equal to 0
+8 +8
add 8 to each side
________________
4x2 = 8
add
____
___
4 4 divide each side by 4
x2 = 2 divide and cancel
_ _
x = √2 x = -√2 take square roots
(iii.) y' = 8x take the derivative
8x = 0 set the derivative equal to 0
___ ___
8
8 divide each side by 8
x = 0 divide and cancel
y(0) = 4(0)2 - 8 replace x with 0
y(0) = -8 multiply combine like terms
results: (0,-8) is the y intercept
_ _
(√2,0)
and (-√2,0) are the x - intercepts
(0,-8) is the relative minimum point
(4.) y = (-3/2)x2 + 4x - 7 here is the problem
(i.)
y = (-3/2)(0)2 + 4(0)
- 7 replace x with 0
y = -7 multiply combine like terms
(ii.) (-3/2)x2 + 4x - 7 =
0 set the function equal to 0
3x2 - 8x + 14 = 0 multiply thru by -2, cancel
b2 - 4ac use the discriminant formula
= (-8)2 - 4(3)(14) make substitutions
=
64 - 168 multiply
= -104 subtract
(iii.) y' = -3x + 4 take the derivative
-3x + 4 = 0 set the derivative equal to 0
3x - 4 = 0 multiply thru by -1
+ 4 +4
add 4 to each side
_______________
3x
= 4 add
____ ___
3 3 divide each side by 3
x = 4/3 cancel
y(4/3) = (-3/2)(4/3)2 + 4(4/3) - 7
replace x with 4/3
y(4/3) = (-3/2)(16/9) + (16/3) - 7
multiply
y(4/3) = (-48/18) + (16/3) - 7
multiply
y(4/3) = (-7/3) + (16/3) - 7
reduce
y(4/3) = -4 combine
like terms
results: (0, -7) is the y intercept
There are no x intercepts
(4/3, -4) is the maximum point
(5.) y = x + (1/x) here is the problem
x = 0 is the vertical asymptote
y = x is the oblique asymptote
y' = 1 - (1/x2) take the derivative of y
1 - (1/x2) = 0 set the derivative equal to 0
+(1/x2) +(1/x2) add this to each side
____________________
1 = 1/x2 add
x2 = 1 multiply each side by x2,
cancel
x = 1
x = -1 take square roots
y = 1 + (1/1) y = (-1) + (1/-1) replace x with 1 & -1
y = 2 y = -2 divide and add
results: (1,2) and (-1,-2) are extreme
points
Here is the graph:
(6.) y = (x2 + 1)/(x + 1) here is the problem
x = -1 this is the vertical asymptote
y = (02 + 1)/(0 + 1) replace x with 0
y = 1 add and divide
(0,1) is the y intercept
There is no x intercept.
-1 |
1 0 1
use synthetic division
___| -1 1
_______________
1 -1
2
result: x - 1 + [2/(x + 1)]
y = x - 1 is the oblique asymptote
y = (x2 + 1)/(x + 1)
y' = [2x(x + 1) - (x2 + 1)]/(x + 1)2 use the quotient rule
y' = (2x2 + 2x - x2 - 1)/(x + 1)2 multiply thru parentheses
y' = (x2 + 2x - 1)/(x + 1)2 combine like terms
x2 + 2x - 1 = 0 set the top
equal to 0
+2 +2
add 2 to each side
________________
x2 + 2x + 1 = 2
add
(x + 1)2 = 2
factor
_ _
x + 1 = √2 x + 1 = -√2 take square roots
-1
-1 -1 -1 subtract
1 from each side
_____________ ____________
_ _
x = -1 + √2
, x = -1 - √2
subtract
_ _ _
y = [(-1 + √2)2
+ 1]/[(-1 + √2)
+ 1] replace x with -1 + √2
_ _
y = (1 - 2√2
+ 2 + 1)/(√2) square the binomial, add -1 & 1
_ _
y = (3 - 2√2)/(√2)
combine like terms
_ _
y = (3√2
- 4)/2 multiply thru by √2
_ _ _
y = [(-1 - √2)2
+ 1]/[(-1 - √2)
+ 1] replace x wiht -1 - √2
_
y = [(1 + 2√2
+ 2) + 1]/(-√2) square the binomial, add -1 & 1
_ _
y = (3 + 2√2)/(-√2)
combine like terms
_ _
y = (-3√2
- 4)/(2) multiply thru by -√2
results: (0,1) is the y - intercept
There is no x intercept
y = x - 1 is the oblique asymptote
_ _ _ _
(-1 +
√2, (3√2 - 4)/2) and (-1 - √2, (-3√2 - 4)/2)
are the extreme points.
(7.) y = x3 + x here is the problem
(i.)
y = (0)3 + (0)
replace x with 0
y = 0 multiply add
(0,0) is the y intercept
(ii.) x3 + x = 0 set the function equal to 0
x(x2 + 1) = 0 factor
x = 0 set this factor equal to 0
(0,0) is the x intercept
(iii.) y' = 3x2 + 1 take the derivative
3x2 + 1 = 0 set the derivative equal to 0
[no solution]
results: (0,0) is the x intercept and
the y intercept
and there are no extreme
points
(8.)(i.) y = x3 - 12x + 10 here is the problem
y = (0)3 - 12(0) +
10 replace x with 0
y = 10 multiply combine like terms
(0,10) is the y intercept
(ii.) y' = 3x2 - 12 take the derivative
3x2 - 12 = 0 set the derivative equal to 0
+ 12
+12 add 12 to each side
________________
3x2 = 12
add
____ ____
3 3 divide each side by 3
x2 = 4 divide and cancel
x = 2 and x = -2
take square roots
y(2) = (2)3 - 12(2) +
10 y(-2) = (-2)3 - 12(-2) +
10
[replace x with 2 and with -2]
y(2) = -6 y(-2) = 26 multiply combine like terms
results: (0,10) is the y intercept
(2,-6) and (-2, 26) are the extremes
(9.) y = x3 - x2 here is the problem
(0,0) is the y intercept
x3 - x2 =
0 set the function equal to 0
x2(x - 1) = 0 factor
x - 1 = 0 set this factor equal to 0
+ 1 +1
add 1 to each side
____________
x = 1 add
(0,0) and (1,0) are the x intercepts
y' = 3x2 - 2x take the derivative
3x2 - 2x = 0 set the derivative equal to 0
x(3x - 2) = 0 factor
x = 0
3x - 2 = 0 set each factor equal
to 0
+ 2
+2 add 2 to each side
______________
3x = 2
add
___ ___
3 3
divide each side by 3
x = 2/3 cancel
y = (2/3)3 - (2/3)2
replace x with 2/3
y = (8/27) - (4/9) multiply
y = (8/27) - (12/27) multiply 4/9 by
3/3
y = -4/27 combine like terms
results: (0,0) and (2/3, -4/27) are the
extremes
(10.) y = x3 - 3x2
- 45x + 25
(0,25) is the y intercept
y' = 3x2 - 6x -
45 take the derivative
3x2 - 6x - 45 = 0 set the derivative equal to 0
___ ___
___ ____
3 3
3 3 divide thru by 3
x2 - 2x - 15 =
0 divide
(x - 5)(x + 3) = 0 factor
x = 5 and x = -3
y(5) = (5)3 - 3(5)2
- 45(5) + 25
y(-3) = (-3)3 - 3(-3)2
- 45(-3) + 25
[replace x with 5][replace x with -3]
y(5) = 125 - 75 - 225 + 25 multiply
y(-3) = -27 - 27 + 135 + 25 multiply
y(5) = -150 y(-3) = 106 combine like terms
(5,-150) and (-3, 106) are the extreme points