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College Algebra
Skill objectives for comprehensive test
LINEAR FUNCTIONS (2.1, 2.2, 2.3, 2.4)
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Understand the quintessential property of being linear:
equal increments in input result in equal increments in output.
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Know that the graph of a (one-variable) linear function is a line.
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Given two different points, calculate the slope of the line through those points.
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Given the slope of a line and a point, write an equation for the line.
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Given the slope of a line and a point, draw a graph of the line.
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Given an equation in point-slope form, slope-intercept form, or interpolation form,
determine the slope of the line.
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Given an equation of a line in any of the three forms, write an equivalent equation in each of the other
two forms.
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Understand that a horizontal line has slope zero, while a vertical line has infinite slope.
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Solve linear equations symbolically.
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Solve linear inequalities symbolically and graphically.
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Use linear functions, equations, and/or inequalities to solve applied problems.
FUNCTION CONCEPT (1.3, 1.4, 1.5, 5.1, 5.2)
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Know that a function returns one output value for each input value.
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Be familiar with function notation and language:
Understand that in the statement "f(x) = x+5",
f is the name of the function,
x represents an input value, and
f(x) represents the output value corresponding to x.
Read f(x) as "f of x".
Given a description (formula, table, graph, or verbal) of a function and a specific input value,
use it to evaluate a function.
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Given symbolic, numerical, graphical, or verbal descriptions for functions f and g, give a corresponding
description for each of these:
f+g, f-g, fg, f /g, fog, gof,...
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Understand that a composition of functions is a function that applies one function after another.
The order in which the functions are applied is explained like this:
"fog" means "do g first, then f ". Read "(fog)(x)" or "f (g(x))" as "f of g of x".
QUADRATIC FUNCTIONS (3.1, 3.2, 3.4)
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Solve a quadratic equation by
taking square roots
factoring
completing the square
using the quadratic formula
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Know that the graph of a quadratic function is a parabola.
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Analyze a quadratic function to find its
y-intercept
[by substituting 0 for x]
x-intercept(s) [by solving f(x) = 0 (because y is zero) ]
vertex
[by completing the square, or use -B/(2A) for x and put into f(x) to find y ]
long-term behavior [by looking at the sign of the x2 coefficient]
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Solve quadratic inequalities graphically and by making a sign chart/table.
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Use quadratic functions, equations, and/or inequalities to solve applied problems, including
projectile motion and revenue problems.
GRAPHS AND TRANSFORMATIONS (mainly in 3.5)
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Given a point in the x-y plane, write the x- and y-coordinates. Given (x, y), plot a point.
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Find each of the following for a given function:
x-intercept(s) [ point(s) of the form (a, 0) ]
y-intercept
[ point of the form (0, b) ]
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Describe what each of the following does to the graph of f(x):
af(x)
[vertical scaling by a factor of a, shrinking if |a| < 1 (if a is "small") ]
[with reflection about the x-axis, if a is negative]
f(x)+d
[shift up/down by d, down if d is negative]
f(x+b)
[shift left/right by b, right if b is negative]
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Given a basic graph and a formula involving one or more of these transformations,
draw the graph of the function. [Example: given p(x), graph -¼p(-x+5) + 3.]
INVERSE FUNCTION CONCEPT (5.2)
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Understand that the inverse function f is the “undo” of the original function f .
Know that the symbol for inverse of a function f is f-1. It does not mean reciprocal.
Given a verbal description of a function f, give one for f-1.
[opposite operation]
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Given a formula for a function f, give a formula for f .
["undo" the steps]
Given a table for a function f, give a table for f-1.
[swap input and output columns]
Given a graph for a function f, give a graph for f-1.
[swap the x and y co-ordinates]
("flip" around the line y = x)
Understand that logarithmic functions are inverses of exponential functions.
Understand that the inverse of a composition of functions is the composition of the inverse functions, but
in reverse order: (f o g)-1 = g-1 o f-1
EXPONENTIAL FUNCTIONS (R.2, R.6, 5.3, 5.6)
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Know the meaning of each of the following or any combination of them:
positive exponents (R.3)
negative exponents (R.3)
exponent of zero (R.3)
fraction exponents (R.7)
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Apply the three basic properties of exponents to evaluate expressions and solve equations.
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Recognize certain powers of small whole numbers:
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Given a numerical or graphical description of a function, tell if the function could be linear, exponential,
or neither.
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Given data, find an exponential function that fits the data.
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Sketch a rough "cartoon" of an exponential function.
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Understand that an asymptote provides a long-term trend for a graph.
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Know that every exponential function can be written in the form f(x) = Cax.
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Describe (in words) the difference between linear and exponential growth.
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Remember that
multiplying by a positive
1
copies
number larger than 1
enlarges
number smaller than 1
reduces
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Know that
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a P% increase is the same as multiplying by (1 + 0.01P)
a P% decrease is the same as multiplying by (1 – 0.01P)
Given an amount with a percent increase/decrease, find the original amount.
Remember that interest rates are usually assumed to be annual.
Know what is meant by compounding (annually, quarterly, monthly, daily).
Given the interest rate, frequency of compounding, length of time, and present value of an investment,
calculate the future value.
Given the other values in the compound interest equation, calculate the present value or the time.
LOGARITHMIC FUNCTIONS (5.4, 5.5, 5.6)
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Know that logarithms are exponents!
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Given a statement in logarithmic form, write an equivalent statement in exponential form.
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Given a statement in exponential form, write an equivalent statement in logarithmic form.
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Know and use the three basic properties of logarithms.
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Use properties of logarithms to solve exponential and logarithmic equations.
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Calculate logarithms of rational numbers (without a calculator).
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Given the interest rate, frequency of compounding, present value, and future value of an investment,
calculate the length of time required.
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Given the interest rate and frequency of compounding of an investment, calculate the doubling time,
tripling time, quadrupling time, etc.
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Solve other application problems involving exponential growth or decay, particularly ones related to
Newton’s law of cooling (or heating) and population growth or decay.
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Sketch a rough "cartoon" of a logarithmic function.
SYSTEMS OF EQUATIONS (6.1, 6.3, 6.4)
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Solve small (two equations) systems of linear equations algebraically.
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Solve small (two equations) systems of nonlinear equations by substitution.
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Understand that a system of linear equations may have one solution, no solution or infinitely many
solutions.
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Understand the difference between a consistent and an inconsistent system.
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Understand the difference between an independent and a dependent system.
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Explain, in detail, the steps of Gaussian elimination with back-substitution.
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Given an application problem, write a system of linear equations for the problem.
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Given a linear system, write the augmented matrix.
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Given an augmented matrix, write the associated linear system.
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Solve an upper-triangular system by using back-substitution.
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Recognize when a matrix corresponds to an inconsistent system.
Know that such a system has no solution.
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Recognize when a matrix corresponds to an dependent system.
Know that such a system may have infinitely many solutions.
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Understand that each variable is a "freedom", while each useful equation is a "restriction".
If there are fewer "restrictions" than "freedoms", the system will have infinitely many solutions.
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In systems that have infinitely many solutions, find three different solutions.
Also, give a complete description of all the solutions using parameters.
MATRIX ARITHMETIC (6.5, 6.6)
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Given a matrix, tell how many rows and how many columns it has.
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Understand that matrix multiplication is “row-by-column”.
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Add or multiply two matrices when it is defined.
POLYNOMIAL FUNCTIONS (R.3, R.4, 4.1, 4.2, 4.3, 4.4)
• Recognize a polynomial function. Know the definition.
• Give its terms, degree, and leading coefficient.
• Add two polynomials.
• Subtract one polynomial from another.
• Multiply two polynomials.
• Divide one polynomial by another, using "long division".
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Divide one polynomial by another, using "synthetic division" where applicable [only if the divisor is of
the form (x-A)].
Analyze a polynomial function to find its
y-intercept
[by substituting 0 for x]
x-intercept(s)
[by solving f(x) = 0 (because y is zero) ]
long-term behavior
[by looking at the sign of the leading coefficient]
Know that the graph of a polynomial function of degree n has at most (n-1) "turns", or n "sections".
Given a polynomial function in complete factored form, sketch a graph of the function
Know how the multiplicity of a "repeated factor" affects the graph of a polynomial function. (4.4)
Solve a polynomial equation by
factoring [including forms A2 −B2= AB A− B , A2 2 ABB2 = AB2 ,
2
2
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A −2 ABB = A−B ,
A3− B3= A−B A2 ABB 2 , A3 B3= AB A2− ABB 2 ] (R.4)
graphing
POLYNOMIAL EQUATIONS (4.3, 4.4, 4.5)
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State and apply the Remainder Theorem [ The remainder when dividing the polynomial f(x) by (x-c) is
f(c). ].
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State and apply the Factor Theorem [ (x-c) is a factor of the polynomial f(x) <==> f(c) = 0 ].
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Know that every polynomial equation of degree n has exactly n solutions, counting complex roots and
"repeats" [because of the Fundamental Theorem of Algebra].
Use synthetic division and/or long division to help you completely factor polynomials and find complex
zeros.
RATIONAL FUNCTIONS (4.6, 4.7)
• Recognize a rational function. Know the definition.
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Know how to find the horizontal asymptote of a rational function.
Know how to find the vertical asymptote(s) of a rational function.
Set up and solve problems involving direct variation and inverse variation.
POLYNOMIAL INEQUALITIES (4.7)
• Solve polynomial inequalities graphically (by sketching a graph).
• Solve polynomial inequalities by making a sign chart/table.
RATIONAL EQUATIONS and INEQUALITIES (4.7)
• Solve rational equations.
• Solve rational inequalities by making a sign chart/table.