I. Rules of radicals, exponents, factoring, products and fractions. Given the right or left hand side of the rule, give the other side. 5 points each

 =

 =

=

=

=

=

=

(a + b)² =

II. Fill in the blank / short answer. 5 points each

What is the complex conjugate of z = 10 - 5i?

For a > 0, write |x| < a without absolute value bars.

For a > 0, write |x| > a without absolute value bars.

Give the definition of a quadratic equation in standard form.

If a quadratic equation has two real roots, what can we say about its discriminant D?

What is the reciprocal of the complex number u = 10 - 5i?

What is the product of the complex numbers (3 + 4i)(3 - 4i)?

If -a is negative, then |a| = _________________.

If a < b and c < 0, then a(-c) ______ b(-c).

If each of a and b is a number, then state the Trichotomy law. 1.__________, 2.__________, or 3._________.

The equation of a vertical line is given by ___________________.

The slope-intercept equation for a straight line is [slope = m, y-intercept = (0,b) ] __________________.

If two non-vertical lines have slopes m₁ and m₂, what is the condition that they be perpendicular in terms of m₁ and m₂?

Find the x-intercepts of the function f(x) by :________________________________________.

A vertical line is one parallel to: _______________________________.

f(x) = log_a(x) is the inverse of ____________________________.

Problems. (5 points)

Find the equation of a vertical line through the point (3, -10).

Find the slope of the line perpendicular to the line y = -3x + 6.

Give the center and radius of the circle: (x - 5)² + (y + 2)² = 64.

Find the x and y intercepts of y = x² + 4x - 4.

Find the vertex of y = x² + 4x - 4.

Evaluate 1 - | 1 - |-1| |.

Simplify (a²)^-3(a³b)²b²(b³)⁴.

Write in exponential form: log₈(4) = 2/3.

Write in exponential form: log₉() = ¼.

Write log₂(8) = 3 in exponential form.

Write 2^x = 6 in logarithmic form.

Write in logarithmic form: 8^-2 = 1/64.

Write in logarithmic form:

Evaluate log₁₄(1) Evaluate log₁₄()

Evaluate log₁₄(196) Evaluate log₁₄(1/14)

Evaluate Evaluate log₁₄()

Evaluate log₂(16). Evaluate log₄(16).

Evaluate log₄(2). Evaluate

Composition of Functions: f(x) = x² + 1, g(x) = 3x - 5, compute fo g(x), go f(x), and fo f(x)

f(x) = x³ + x² - 6, g(x) = x + 1/x, compute fo g(x), go f(x), and fo f(x)

IV. Plots. Answer on separate graph paper. (10 points each)

Draw the circle x² + y ² = 9 Sketch the graph of y = x² + 4x - 4.

Sketch the graph of y = log₃(x). Graph y = 3^-x.

V. Problems. Show all work. Find all solutions. 10 points each. Answer on separate paper.

Find the domain of

Find the equation of a line perpendicular to the line y = -3x + 4 passing through the point (4,6).

Find the domain of the function .

If f(x) = 6 - 4x, compute f^-1(x).

Solve for x: 2^{1 - x} = 3

Solve for x: log(x) + log(x - 1) = log(4x).

Rewrite as single log: log(12) + (1/2)log(7) - log(3).

Rewrite as a single log: log(a + b) + log(a - b) - 2log(c).

Expand completely using the rules of logarithms only:
Find two positive numbers whose sum is 100 and the sum of whose squares is a minimum.

The height (y)of a ball thrown by a child is given by , where x is the horizontal distance in feet from where the ball was thrown. Sketch the path of the ball. How high was the ball when it left the child’s hand? What was the maximum height of the ball? How far from the child did the ball strike the ground.

The half-life of Radium-226 is 1590 years. If a sample has a mass of 150mg, find a formula for the mass that remains after t years. Find the mass that remains after 1000 years.

A bacteria culture starts with 8000 bacteria and doubles every 20 minutes. When will the population reach a level of 32,000.