Download Pre-Calculus 20 Final Review

Pre-Calculus 20 Final Review
June 2015
Name: ____________________________
P20.1 Demonstrate understanding of the absolute value of real numbers and equations and
functions involving the absolute value of linear and quadratic functions
Level 2
1. Evaluate.
Level 3
2. Solve the following equation, and verify .Label the extraneous solution if possible. (Level 3)
Case 1:
Case 2:
|7𝑥 − 3| = 𝑥 + 1
Verify:
Verify:
Level 4
3. A machine fills containers with 64 ounces of oatmeal. After the containers are filled, another
machine weighs them. If the container’s weight differs from the desired 64-ounces weight by
more than ±0.8 ounces, the container is rejected. Write an absolute value equation that can be
used to find the heaviest and lightest acceptable weights for the container.
P20.2 Expand and demonstrate understanding of radicals with numerical and variable radicands
Level 2
1.Express each entire radical as a mixed radical in simplest form.
2. Write each mixed radical as an entire radical.
Level 3
3.Solve the following radical equations, and verify. Label the extraneous solution if possible.
Level 4
4. The speed, V, in feet per second of outlow of a liquid from an orfice is given by the formula
𝑉 = 8√ℎ , where h is the height, in eet, of the liquid above the opening. How high, to the
nearest hundereth of a foot, is a liquid above an office if the velocity of outflow is 100 ft/s?
P20.3 Expand and demonstrate understanding of rational expressions and equations
Level 2
1. Simplify and state the non-permissible values for the variables.
a)
b)
Level 3
2. Solve the following rational equation, and verify. Label the extraneous solution if possible.
Level 4
3. Three numbers are related to one another such a way that the second number is 8 larger
than the first and the third number is three times as large as the first. Find the numbers if the
sum of three-quarters of the first number, two-thirds of the second number and one-sixth of
the third number is 36.
P20.4 Expand and demonstrate understanding of the primary trigonometric ratios including the
use of reference angles and the determination of exact values for trigonometric ratios
Level 2
1. Sketch each angle in standard position. State which quadrant the angle terminates in and the
measure of the reference angle.
a) 20°
quadrant _________________________
θR= ____________________
b) 330°
quadrant _________________________
θR= ____________________
Level 3
2. Solve for θ satisfy the equation for 0°<θ≤360°?
1
a)cos 𝜃 = 0.3420
b) sin 𝜃 = − 2
Level 4
3. A guy wire supporting a radio tower is positioned 135 feet up the tower. It forms a 45˚ angle with the
ground. About how long is the wire, using the exact length? (Level 4)
P20.5 Demonstrate understanding of the cosine law and sine law, including the ambiguous case
Level 2
1. Find the indicated angle or side length. Round your answer to one decimal place.
a)
b)
c)
Level 3
2. In ∆𝐴𝐵𝐶, where ∠𝐴 = 29°, 𝑎 = 17, and 𝑏 = 18. Find angle(s) ∠𝐶.
Level 4
3. A 5m flag pole is not standing up straight. There is a wire attached to the top of the pole and
anchored in the ground. The wire is 5.17m long. The wire makes a 52° angle with the ground.
What angle does the flag pole make with the wire? Round your answer to the nearest degree.
P20.6 Expand and demonstrate understanding of factoring polynomial expressions
Level 2
1. Factor the following:
1
a) 𝑎2 − 49𝑏 2
b) 𝑥 2 − 5𝑥 − 6
c) 4𝑥 2 − 36𝑦 2
d) 6𝑥 2 + 5𝑥 − 6
e) 𝑥 2 − 3𝑥 − 4
f) 2𝑥 2 + 7𝑥 + 3
9
Level 3
2. Factor the following:
Level 4
3. One side of an envelope is 5 inches longer than the other side. The area of the envelope is
110 in2. Determine the dimensions of the envelope.
P20.7 Demonstrate understanding of quadratic functions of the form 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 and
their graphs
Level 2
1. a) Graph the following quadratic function:
𝑦 = −6(𝑥 − 4)2 + 3
b) For the graph state the following:
direction of opening: __________________________________________________
domain: _____________________________________________________________
range: _______________________________________________________________
axis of symmetry: ______________________________________________________
x-intercepts: _________________________________________________________
y-intercepts: __________________________________________________________
Level 3
2. a) Convert the following quadratic functions from standard form to vertex graphing form.
b)State the vertex.
Level 4
3. A basketball is thrown into the air from ground level and its path ia a parabola. It reaches a
maximum height of 8 m and lands 20m from where it was thrown. Determine an eqation that
models the path of the ball.
P20.8 Demonstrate understanding of quadratic equations
Level 2
1. Solve using any method.
Level 3
2. Solve by completing the square. Express answers to two decimal places.
Level 4
3. List the one advantage and one disadvantage to each way of solving a quadratic equation.
Factoring:
Completing the square
Quadratic Formula:
P20.10 Demonstrate understanding of arithmetic and geometric sequences and series
Level 2
1. Identify whether the following sequences are arithmetic, geometric, or neither.
Level 3
2. Find the sum of the first 10 terms of the arithmetic series.
6+3+0+⋯
Level 4
3. The length of an initial swing of a pendulum is 120cm. Each successive swing is 0.90 times the
length of the previous swing. If this process continues forever, how far will the pendulum swing
in total?
P20.11 Demonstrate understanding of reciprocal functions
Level 2
1
1. Use the graph of y= f(x) given here to graph 𝑦 = 𝑓(𝑥) on the same axis. Show the vertical
asymptotes on your graph.
a)
b)
Level 3
1
2. Sketch the graph 𝑦 = 2𝑥+2 . Label the asymptote(s).
Level 4
3. Consider this graph of a reciprocal function 𝑦 =
graph of the original function 𝑦 = 𝑓(𝑥).
1
. On the same set of axes, sketch the
𝑓(𝑥)
Formulas
𝑎
𝑏
𝑐
=
=
sin 𝐴 sin 𝐵 sin 𝐶
𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶
−𝑏 ± √𝑏 2 − 4𝑎𝑐
𝑥=
2𝑎
Arithmetic
𝑡𝑛 = 𝑡1 + (𝑛 − 1)𝑑
𝑛
𝑆𝑛 = 2 [2𝑡1 + (𝑛 − 1)𝑑]
𝑛
𝑆𝑛 = 2 (𝑡1 + 𝑡𝑛 )
Geometric
𝑡𝑛 = 𝑡1 (𝑟)𝑛−1
𝑆𝑛 =
𝑡1 (𝑟 𝑛 − 1)
(𝑟 − 1)
𝑆𝑛 =
𝑟𝑡𝑛 − 𝑡1
𝑟−1
𝑡
1
𝑆∞ = (1−𝑟)