Name__________________________ Period_____ Sec8-6 Notes: Mixture Problems & Review
HW: Sec8-6 #5, 9 and all the problems at the end of the notes
REVIEW: Solve proportions by cross multiplying
(1)
2
6

x  3 4x  5
(2) 0.2 
3.4
4x  5
You Try 0.3 
2.4
2x  4
MIXTURE PROBLEMS: Rational expressions are very useful in mixture problems.
Example A: Suppose two quarts of chocolate milk contains
2% chocolate syrup and 98% milk. You want to add more
chocolate syrup so that your chocolate milk contains 5%
chocolate syrup. How much chocolate syrup do you need
to add?
+
=
Total Volume:
Let x be the amount of syrup you add.
2 quarts
+ x quarts
2% chocolate
syrup
Amount of chocolate syrup: (0.02)(2)
+ x
You want the percentage chocolate in the mixture, P, to be 5%.
P  5% 
Solve this proportion for x:
5
Amount of chocolate syrup in mixture (0.02)2  x


100
Total volume in mixture
2 x
Example B: Suppose you have 100 mL of a solution that is 30%
acid and 70% water. How many mL of acid do you need to add
to make a solution that is 60% acid?
Solution: Let x be the amount of acid you add
= 2 + x quarts
5% chocolate
Total Volume:
5
0.04  x

100
2 x
+
100 ml +
30% acid
Amount of acid: (0.30)(100) +
=
x ml
acid
= 100 + x ml
60% acid
x
Solve this proportion for x:
You want the percentage acid in the mixture, P, to be 60%.
P  60% 
60 Amount of acid in mixture (0.30)100  x 30  x



100
Total volume in mixture
100  x
100  x
60
30  x

100 100  x
Name__________________________ Period_____ Sec8-6 Notes: Mixture Problems & Review
HW: Sec8-6 #5, 9 and this worksheet.
(1) Suppose you have 70 ml of an acid solution that is 4% acid and 96% water. You want to add acid to
the solution so that you have 10% acid. How much acid do you need to add?
#2-5 No calculator: Review how to graph both horizontal and vertical parabolas.
Label the vertex, both stretch factors and on other point on the graph.
2
(3) y  2( x  1)2  5
 y4
(2) 
  x 1
 3 
 y 3
 x5
(5) 
  

 10 
 20 
 y  4  x 5
(4) 
 

 6   4 
2
2
#6-9 Review how to find the focus and directrix. Sketch the given information.
(Sec8-3 notes)
(6) Find the equation of the directrix if the
focus is at (1,7) and the vertex is at (1,3)
(7) Find the equation of the directrix if the
focus is at (1,7) and the vertex is at (9,7)
(8) Find the coordinate of the vertex is the if
the focus is at (1,7) and the directrix is
y=5
(9) Find the coordinate of the vertex is the if
the focus is at (1,7) and the directrix is
x=9
#10-13 No calculator
Review how to find the focus and directrix when you are given an equation of a parabola.
Sketch the graph. On your graph label the vertex, directrix and focus
Recall that f is the distance from the vertex to the focus and also from the vertex to the directrix.
Look at the last page of Sec8-3 notes to find the relationship between f and the stretch factors.
1
( x  5) 2
16
(10)
 y7

  ( x  1)
 5 
(11)
y
(12)
 y 7

  ( x  1)
 5 
(13)
y
2
2
1
( x  5) 2
16
#14-17 Change each equation into the standard form of either: a circle, an ellipse, a vertical
parabola or a horizontal parabola. You will need to complete the square either with one or
both variables. Sketch the graph. Include all the important features: Center of the
circle or center of the ellipse, vertex of parabola along with all the vertical and
horizontal stretch factors (Sec8-5 notes).
(14)
2 x 2  12 x  y  11  0
(15)
y 2  2 y  4 x  19  0
(16)
x 2  8 x  y 2  10 y  37  0
(17)
25 x 2  150 x  4 y 2  32 y  189  0
#18-21 Rewrite in factored form by factoring the numerator and denominator completely.
Be prepared to share your answers in class.
(18)
x2  6 x  5
y
x 1
(20)
y
x 2  9 x  14
x2  6x  8
(19)
y
x 3
x  2 x  15
(21)
y
x 2  2 x  35
x 2  4 x  21
2
#22-23 No calculator
1
and graph.
x
Identify both asymptotes on the graph and the vertical stretch factor, then find the domain and range.
Use long division to rewrite rational functions as transformations of f ( x ) 
(22)
Divide
10 x  32
y
x3
x  3 10 x  32
Rewrite as transformation of
1
f ( x) 
x
y  _______________
Asymptotes: x = _____
& y = ____
Domain:
What is the vertical
stretch factor for this
graph?
(23)
Range:
Sketch asymptotes using dashed lines.
Label one point on the graph that shows
the vertical stretch factor.
Divide
4 x  23
y
x5
x  5 4 x  23
Rewrite as transformation of
1
f ( x) 
x
y  _______________
Asymptotes: x = _____ &
y = ____
Domain:
What is the vertical
stretch factor of this
graph?
Range:
Sketch asymptotes using dashed lines.
Label one point on the graph that shows
the vertical stretch factor.