Algebra Class: Functions and Polynomials
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Questions and Answers

What is the linear function that expresses temperature T in relation to height h?

  • T = -20h + 10
  • T = 10h - 20
  • T = 10h + 20
  • T = -10h + 20 (correct)

    • The slope of the temperature function represents an increase in temperature with increasing height.

    False

    What is the temperature at a height of 2.5 km according to the derived linear function?

    -5°C

    A polynomial of degree 1 is a ______ function.

    <p>linear</p> Signup and view all the answers

    Match the following polynomial degrees with their corresponding function type:

    <p>Degree 1 = Linear Function Degree 2 = Quadratic Function Degree 3 = Cubic Function Degree 4 = Quartic Function</p> Signup and view all the answers

    What does the negative slope of -10°C/km indicate in the temperature function?

    <p>Temperature decreases as height increases.</p> Signup and view all the answers

    The domain of any polynomial function includes all real numbers.

    <p>True</p> Signup and view all the answers

    What is the leading coefficient of the polynomial P(x) = -2x^3 + 3x^2 - x + 5?

    <p>-2</p> Signup and view all the answers

    What is the absolute value of -5?

    <p>5</p> Signup and view all the answers

    All absolute values are greater than or equal to zero.

    <p>True</p> Signup and view all the answers

    What type of function is f(x) = |x|?

    <p>Even function</p> Signup and view all the answers

    How can the limit notation 'lim as x approaches a of f(x) = ∞' be alternatively expressed?

    <p>f(x) becomes infinite as x approaches a</p> Signup and view all the answers

    The cost C(w) of mailing a large envelope is $____ when the weight is between 1 and 2 pounds.

    <p>1.00</p> Signup and view all the answers

    Match the following weights with their corresponding mailing costs:

    <p>0 &lt; w ≤ 1 = $0.83 1 &lt; w ≤ 2 = $1.00 2 &lt; w ≤ 3 = $1.17 3 &lt; w ≤ 4 = $1.34</p> Signup and view all the answers

    The symbol ∞ is considered a number in limit notation.

    <p>False</p> Signup and view all the answers

    What does 'lim as x approaches a of f(x) = -∞' indicate about the behavior of the function f(x)?

    <p>f(x) decreases without bound as x approaches a</p> Signup and view all the answers

    Which of the following represents the piecewise definition of the absolute value function?

    <p>x if x ≥ 0; –x if x &lt; 0</p> Signup and view all the answers

    The function f(x) = x² is an odd function.

    <p>False</p> Signup and view all the answers

    The vertical asymptotes of __________ are found where cos x = 0.

    <p>tan x</p> Signup and view all the answers

    Match the types of limits with their descriptions:

    <p>lim as x approaches a of f(x) = ∞ = f(x) increases without bound as x approaches a lim as x approaches a of f(x) = -∞ = f(x) decreases without bound as x approaches a lim as x approaches a- of f(x) = considers values of x that are less than a lim as x approaches a+ of f(x) = considers values of x that are greater than a</p> Signup and view all the answers

    What is the geometric significance of an even function?

    <p>Symmetry with respect to the y-axis</p> Signup and view all the answers

    What does the graph of y = -f(x) represent?

    <p>The graph is reflected about the x-axis</p> Signup and view all the answers

    A transformation where the function is multiplied by a positive number greater than 1 results in a vertical stretch.

    <p>True</p> Signup and view all the answers

    What happens to the graph of a function when we apply the absolute value transformation?

    <p>The part of the graph below the x-axis is reflected about the x-axis.</p> Signup and view all the answers

    The graph of y = 2 cos x is obtained by stretching the graph of y = cos x vertically by a factor of ______.

    <p>2</p> Signup and view all the answers

    Match the following transformations with their effects on the graph:

    <p>y = f(x + 2) = Shifted 2 units left y = f(x - 2) = Shifted 2 units right y = -f(x) = Reflected about the x-axis y = |f(x)| = Reflected below the x-axis</p> Signup and view all the answers

    How does the transformation y = f(x - 2) affect the graph of f?

    <p>The graph shifts 2 units right</p> Signup and view all the answers

    The graph of y = f(x) shifts downward if you add a positive constant to f(x).

    <p>False</p> Signup and view all the answers

    What effect does reflecting a graph about the y-axis have on the coordinates of points?

    <p>It changes the sign of the x-coordinates.</p> Signup and view all the answers

    Which limit laws are helpful in calculating limits?

    <p>All of the above</p> Signup and view all the answers

    The one-sided limits must be equal for a two-sided limit to exist.

    <p>True</p> Signup and view all the answers

    What is the primary condition for a function to be continuous at a point a?

    <p>The function must have the Direct Substitution Property.</p> Signup and view all the answers

    The _________ Theorem states that if g(x) is squeezed between f(x) and h(x), then g must have the same limit as both f and h.

    <p>Squeeze</p> Signup and view all the answers

    Match the following functions with their properties:

    <p>Continuous Function = Direct Substitution Property Greatest Integer Function = Does not exist at certain points Squeeze Theorem = One-sided limits equal Root Law = Applies to roots of functions</p> Signup and view all the answers

    Which statement about the greatest integer function is true?

    <p>It does not exist at certain points.</p> Signup and view all the answers

    The Direct Substitution Property can be used for all functions when calculating limits.

    <p>False</p> Signup and view all the answers

    What is indicated by the notation 'lim x → a f(x)'?

    <p>It indicates the limit of f(x) as x approaches a.</p> Signup and view all the answers

    What is the derivative of the function $f(x) = 5x^6$?

    <p>$30x^5$</p> Signup and view all the answers

    The derivative of a sum of functions is equal to the sum of their derivatives.

    <p>True</p> Signup and view all the answers

    What rule is used to derive the derivative of a product of two functions?

    <p>Product Rule</p> Signup and view all the answers

    The derivative of $x^4$ is ______.

    <p>4x^3</p> Signup and view all the answers

    Which of the following is the result of the derivative $g(x) = 2x^3 + 5x^2 - x$?

    <p>$6x^2 + 10x - 1$</p> Signup and view all the answers

    The Constant Multiple Rule states that the derivative of a constant times a function is equal to the function's derivative multiplied by the constant.

    <p>True</p> Signup and view all the answers

    What is the derivative of the function $y = -3x^4$?

    <p>-12x^3</p> Signup and view all the answers

    Match the following derivatives with their corresponding functions:

    <p>f(x) = x^5 = 5x^4 y(x) = 7x^3 = 21x^2 g(x) = -2x^2 = -4x h(x) = 4x^6 = 24x^5</p> Signup and view all the answers

    Study Notes

    Precalculus Review

    • This document is a precalculus review, providing key algebraic concepts.
    • Real numbers, denoted ℝ, encompass all numbers on the number line (positive, negative, zero, fractions, etc.).
    • Rational numbers (ℚ) can be expressed as fractions (e.g., -0.5).
    • Integers (ℤ) are whole numbers (e.g., -4, -3, -2, -1, 0, 1, 2).
    • Natural numbers (ℕ) are positive integers (e.g., 1, 2, 3).
    • Infinity (∞) represents values that become arbitrarily large. Negative infinity (-∞) represents values tending to negative infinity.
    • Intervals are collections of real numbers between two fixed numbers (endpoints may or may not be included).
    • Specific interval notations are defined, with explanations of open, closed, half-open intervals and unbounded intervals.
    • The absolute value function|x| returns the distance of x from 0.
    • Key algebraic formulas for squares and products of binomials are presented.
    • Solving quadratic equations, where appropriate to solving using factoring is reviewed.
    • Fractions and rational expressions, including operations and simplification techniques.
    • Functions defined piecewise are reviewed (with solid/hollow dots).
    • Functions are explained verbally, numerically, graphically, or algebraically.
    • The concepts of domain, range, and graph of function.
    • Definition of a function is a rule that assigns each element in a set D (the domain) to exactly one element in a set E (the range).

    1.1 Algebra

    • Review of specific types of intervals in mathematical context.
    • Review of absolute value function properties and how to apply them to various calculations.
    • Review of algebra formulas for multiplying and factoring various polynomials.
    • Thorough review of how to solve quadratic equations.

    1.2 Functions and Graphs

    • Review of different ways of representing a function.
    • Explanations about domain, range, and how to read information from function graphs.
    • Definition of function is review and is stated precisely.
    • Explanations on how to determine domain and range from graph.

    1.3 Linear Functions

    • Linear Function definitions, including slope and y-intercept formulas/definitions.
    • Various forms of equations of lines, including the general form, vertical lines, and non-vertical lines.
    • Review of parallel and perpendicular lines.

    1.4 Polynomials

    • Definition of a polynomial.
    • Explanation on coefficients, leading coefficients, and the constant term of a polynomial.
    • Polynomial degree definition and example.
    • Domain of a polynomial is always all real numbers, if leading coefficient is not zero.
    • Review of graphs of polynomial functions and their degrees.

    1.5 Power Functions

    • Definitions of power functions and how a is evaluated for a= n or if a=1n.
    • Explanation on various types of exponents and how to define the numerical value of the corresponding function value.
    • Understanding how to evaluate various numerical power functions.

    1.6 Trigonometric Functions

    • Review of radians and degrees in mathematical context.
    • Detailed definitions of all six trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant).
    • How these are calculated and evaluated.
    • Special values of trigonometric functions for common angles.
    • Trigonometric identities reviewed.
    • Graphs of the sine, cosine, and tangent functions.

    1.7 Exponential Functions

    • Definition of exponential function (and its base).
    • Review of the number e.
    • Graphs of exponential functions of the form y = ax where a is a constant.

    1.8 Logarithmic Functions

    • Definition of logarithmic function (and its base).
    • Review of the number e and its inverse function relation with exponential function.
    • Definition of common and natural logarithmic functions, and graphs of such functions with various bases (e.g. 2, 3, 5 , and 10).
    • Review basic logarithmic formulas.

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    Description

    Test your knowledge on linear functions, polynomial degrees, and absolute values in this algebra class quiz. Explore key concepts like temperature relationship with height, leading coefficients, and limit notation through various questions. Perfect for reinforcing your understanding of function types and properties!

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