Precalculus PreAP
_____________________________
Summer Assignment
Name/Class Period
Show all work on your own paper.
a
3a
1) Verify using algebra that the points (0, 0), (a, 0), ( ,
) are the
2
2
vertices of an equilateral triangle. Then show that the midpoints of
the three sides are the vertices of a second equilateral triangle.
2) To test the graph of an equation for any symmetries, apply each of
the following tests:
a) X-axis symmetry Replace y by – y in the equation. If an
equivalent equation results, the graph is symmetric with respect
to the x-axis.
b) Y-axis symmetry Replace x by – x in the equation. If an
equivalent equation results, the graph is symmetric with respect
to the y-axis.
c) Origin symmetry Replace x by (–x) and y by (–y) in the
equation. If an equivalent equation results, the graph is
symmetric with respect to the origin.
Test y =
4x2
for any symmetries. Show your work.
x2 + 1
3) Solve each equation.
4
18
a) - 5 =
b) x3 + x2 – 4x – 4 = 0
y
2y
1
d) Solve T = gt2 for t.
3
e) Solve
c)
2
3
-8
+
=
x+2 x-1 5
x+2
=k
y2
for y.
4) For the quadratic function y = - 2x2 – 12x -10,
a) Transform the equation to the form y – k = a(x – h)2.
b) Find the vertex.
c) Find the x- and y-intercepts.
d) Sketch the graph.
e) Shade the region on the graph corresponding to the solution set of
the quadratic inequality y > - 2x2 – 12x -10.
5) When you compress air in a tire pump, the air heats up. A diesel
engine works on the same principle. Air in the cylinders is compressed
until it is hot enough to ignite the diesel fuel. As the air is
compressed, its absolute temperature varies inversely with 0.4 power
of its volume.
a) As the air starts being compressed, it is at “room temperature” of
300oK, and occupies 60 cu. cm. volume. Write an equation
expressing absolute temperature in terms of volume.
b) What is the temperature when the air is compressed to 4 cu. cm.?
c) The minimum temperature needed to make diesel fuel ignite is
750oK. To what volume must the air be compressed to reach this
temperature?
6) When you dive under water, nitrogen from the air you breathe
dissolves in your blood. According to Henry’s Law, the concentration
of nitrogen in your blood varies linearly with the depth at which you
are swimming. At 20 feet, the concentration is about 190 mg of
nitrogen per liter of blood. At 30 feet, the concentration is about
225 mg per liter.
a) Write the particular equation expressing nitrogen
concentration in terms of depth.
b) Predict the nitrogen concentration at 100 feet.
c) If too much nitrogen dissolves in your blood, you will get
“nitrogen narcosis”. Assume that this narcosis sets in when the
concentration reaches 680 mg per liter. How deep can you
safely dive?
d) According to your mathematical model, how much nitrogen is
normally dissolved in your blood when you are out of the water?
What part of the model tells you this?
7) The Richter Scale measures the magnitude R of an earthquake. It is
defined by the formula {r = 0.67 log (0.37E) + 1.46} where E is the
energy (in kilowatt-hours) released by the quake. The 1933 quake in
Japan measured 8.9 on the Richter Scale. In scientific notation (with
3 significant digits), how much energy was released?
8) Given: f(x) = x2 – 4, x > 0
a) State the range of the function.
b) On graph paper, sketch f(x) over its domain. Label the graph
f(x).
c) On the same set of axes, sketch f-1(x), the inverse of f(x).
Label the graph f-1(x).
d) Write an equation for f-1(x).
e) Find f(f-1(5)).
9) Which logarithmic equation is equivalent to Lm = E ?
a) logm E = L
c) logE L = m
b) logL E = m
d) logE m = L