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 (B)

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. (B)</p> Signup and view all the answers

The domain of any polynomial function includes all real numbers.

<p>True (A)</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 (A)</p> Signup and view all the answers

All absolute values are greater than or equal to zero.

<p>True (A)</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 (B)</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 (B)</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 (D)</p> Signup and view all the answers

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

<p>False (B)</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 (C)</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 (A)</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 (B)</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 (B)</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 (D)</p> Signup and view all the answers

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

<p>True (A)</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. (A)</p> Signup and view all the answers

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

<p>False (B)</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>$30x^5$ (C)</p> Signup and view all the answers

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

<p>True (A)</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>$6x^2 + 10x - 1$ (D)</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 (A)</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

Flashcards

Absolute Value

The distance of a number from zero on the number line.

Piecewise Defined Function

A function described by different formulas for different parts of its domain.

Even Function

A function where f(-x) = f(x) for all x in its domain. Its graph is symmetric across the y-axis.

Odd Function

A function where f(-x) = -f(x) for all x in its domain. Its graph is symmetric about the origin.

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f(x) = |x|

The absolute value function, defined as |x| = x if x >= 0 and |x| = -x if x < 0.

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Step Function

A function that increases or decreases in steps. Its graph looks like a series of horizontal lines.

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|| (Vertical Bars)

The symbol used to represent the absolute value of a number.

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C(w) - Cost of Mailing a Large Envelope

A function that represents the cost of mailing a large envelope, where the cost depends on the weight of the envelope.

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Reflection About the x-axis

Reflecting a function's graph about the x-axis changes the sign of the y-coordinate for every point on the graph. Think of folding the graph along the x-axis.

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Vertical Stretching/Shrinking

Multiplying the y-coordinate of every point on a graph by a constant 'c' results in either stretching or shrinking the graph vertically, depending on the value of 'c'. If 'c' is greater than 1, the graph is stretched. If 'c' is between 0 and 1, the graph is shrunk.

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Graph of y = |f(x)|

The graph of 'y = |f(x)|' is obtained by reflecting the part of the graph of 'y = f(x)' that lies below the x-axis about the x-axis. The part of the graph that lies above the x-axis remains the same.

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Limit of a Function

The limit of a function f(x) as x approaches a is the value that f(x) gets closer and closer to as x gets closer and closer to a, without actually reaching 'a'. It indicates the behavior of the function near a point.

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Linear Temperature Model

A mathematical relationship that represents the change in temperature with respect to height, where the temperature decreases linearly as the height increases.

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Slope of the Temperature Function

The rate at which the temperature changes with respect to height. It is the slope of the linear function.

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Polynomial Function

A mathematical function that describes a curved line, determined by its coefficients, which can be used to represent various relationships.

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Degree of a Polynomial

The highest power of the variable in a polynomial function determines its degree. For example, a quadratic function has a degree of 2.

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Quadratic Function

A specific type of polynomial function with a degree of 2, represented by the equation P(x) = ax^2 + bx + c.

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Parabola

The graph of a quadratic function, which is always a symmetric, U-shaped curve. It opens upwards if the leading coefficient is positive and downwards if negative.

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Empirical Model

A mathematical model that is based on observed data and seeks to capture the general trend of the data points.

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Y-intercept

A mathematical concept used to represent a linear relationship where the y-intercept is the value of the function when the independent variable is zero.

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f(x)  as x a

A way to represent the limit of a function f(x) as x approaches a when the function grows infinitely large.

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Vertical Asymtote

A line that the graph of a function approaches as the input approaches a certain value, but never actually touches.

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limxa f(x) = 

A way to express that a function f(x) grows infinitely large (either positive or negative) as x approaches a particular value.

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limxa- f(x) = L

The limit of f(x) as x approaches 'a' from the left side (values of x less than 'a').

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limxa+ f(x) = L

The limit of f(x) as x approaches 'a' from the right side (values of x greater than 'a').

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limxa f(x) = –

A specific type of limit where the function's output becomes infinitely large and negative as x approaches a certain value.

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Two-Sided Limit

The limit of a function exists if and only if both the left-hand and right-hand limits exist and are equal.

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One-Sided Limit

The limit of f(x) as x approaches a from either the left or the right side of 'a' is infinity.

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Continuous function

A function where the limit exists and equals the function value at the point. In simple terms, you can find the limit by simply plugging in the number.

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Constant Multiple Rule

The derivative of a constant times a function is equal to the constant multiplied by the derivative of the function.

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Sum Rule

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

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Left-hand limit

Calculates the limit of a function as x approaches a number 'a', but from the smaller side (values less than 'a').

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Difference Rule

The derivative of a difference of functions is equal to the difference of the derivatives of each function.

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Right-hand limit

Calculates the limit of a function as x approaches a number 'a', but from the larger side (values greater than 'a').

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Product Rule

The derivative of a product of two functions is the first function multiplied by the derivative of the second function, plus the second function multiplied by the derivative of the first function.

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Derivative of a Constant

The derivative of a constant is always zero.

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Limit Theorem

A theorem stating that a two-sided limit exists only if both the left-hand and right-hand limits exist and are equal.

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Power Rule

The derivative of x raised to a power (n) is equal to n times x raised to the power (n-1).

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Differentiating Complex Functions

Any function can be broken down into simpler functions, and its derivative can be calculated by applying the appropriate rules for each simpler function.

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Squeeze Theorem

A function squeezed between two other functions with the same limit, also has the same limit. Think of a sandwich.

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Direct Substitution Property

A function that can be evaluated by simply plugging in the value of x. It does not have any holes or breaks.

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Derivative as Rate of Change

The derivative is a measure of the instantaneous rate of change of a function. It tells us how fast a function is changing at a specific point.

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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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