AP Precalculus Flash Card Highlights 2024

Function Transformations

• Vertical Dilation

  • The parameter a in the function $g(x) = a f(b(x - h)) + k$ determines how much the graph stretches or compresses vertically.

    • Dilation occurs by a factor of |a|:

      • If |a| > 1, the graph stretches vertically, which means that points on the graph move away from the x-axis.

      • If 0 < |a| < 1, the graph compresses vertically, bringing points closer to the x-axis.

      • Reflection: If $a < 0$, the graph will reflect over the x-axis, flipping it upside down.

• Horizontal Translation

  • The parameter h affects the horizontal position of the graph.

    • When $x = h + c$, the graph moves to the left by c units, while for $x = h - c$, it moves to the right by c units.

      • This shift does not alter the shape of the graph, only its position along the x-axis.

• Vertical Translation

  • The parameter k determines the vertical position of the graph.

    • If $k > 0$, the graph shifts upward by k units, and if $k < 0$, it shifts downward by |k| units.

      • The overall shape and orientation of the graph remain unchanged, only its vertical location changes.

• Horizontal Dilation

  • The parameter b influences the horizontal stretching or compressing of the graph:

    • Dilation happens by a factor of $1/b$:

      • If |b| > 1, the graph compresses horizontally, bringing points closer together along the x-axis.

      • If 0 < |b| < 1, the graph stretches horizontally, extending farther apart.

      • Reflection: If $b < 0$, the graph will reflect over the y-axis, flipping its orientation.

Average Rate of Change and Behavior

• Average Rate of Change

  • This metric is calculated between two specific points $(a, f(a))$ and $(b, f(b))$ and is expressed mathematically as: \text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}.

    • It represents the slope of the secant line connecting these two points on the function's graph.

    • A positive average rate indicates that the function is increasing over that interval, while a negative average rate indicates it is decreasing.

• Function Behavior

  • Positive Function: When the y-coordinates of the graph are above the x-axis, indicating that for all x in this interval, $f(x) > 0$.

  • Negative Function: When the y-coordinates are below the x-axis, indicating $f(x) < 0$.

  • Increasing Function: The graph rises as x increases; numerically, this means $f(b) > f(a)$ for $b > a$.

  • Decreasing Function: The graph falls as x increases; numerically, this means $f(b) < f(a)$ for $b > a$.

Inflection and Concavity

• Point of Inflection:

  • This is the point on the graph where the concavity changes, meaning the graph switches from being concave up to concave down or vice versa.

    • Inflection points can be found by determining where the second derivative of the function changes sign.

• Concavity and Rate of Change

  • Concave Up:

    • Indicates that the slope of the tangent line is increasing, which suggests that the rate of change of the function is accelerating positively.

    • Thus, if you pick two points on a concave-up section, the average rate of change between those points will be less than the instantaneous rate of change at any point between them.

  • Concave Down:

    • Indicates that the slope of the tangent line is decreasing, which means the function is slowing down in a positive direction or speeding up in a negative direction.

    • In this case, the average rate of change will be greater than the instantaneous rate between those two points.

Zeros and Function Characteristics

• Odd Multiplicity (c is odd)

  • If $c$ is a zero of the function, it means $f(c) = 0$.

    • The graph crosses the x-axis at $x = c$, which signifies a change of sign of the function.

• Even Multiplicity (c is even)

  • If $c$ is a zero, it means that the graph will touch the x-axis and turn around at $x = c$.

    • This reflects a zero that does not change the sign of the function; it remains the same on either side of the zero.

• Even Function:

  • Defined by the property $f(-x) = f(x)$, indicating symmetry about the y-axis; for any point on the graph, there is a corresponding point directly opposite across the y-axis.

• Odd Function:

  • Defined by the property $f(-x) = -f(x)$; this indicates symmetry about the origin, meaning if you reflect a point across both axes, you get another point on the graph.

End Behavior and Asymptotes

• Notation for End Behavior:

  • For understanding how a function behaves as it approaches positive or negative infinity, we use:

    • As inputs decrease without bound:
      \text{lim}\_{x \to -\infty} f(x)

    • As inputs increase without bound:
      \text{lim}\_{x \to +\infty} f(x)

  • This notation is useful to describe the horizontal asymptotes or the overall trend of the function.

• Horizontal Asymptotes (H.A.):

  • These are determined by comparing the degrees of the numerator and denominator of a rational function.

    • If degree of numerator $n < m$ (degree of denominator), then the H.A. is at $y = 0$.

    • If $n = m$, the H.A. is at y = \frac{\text{L.C. of numerator}}{\text{L.C. of denominator}}.

    • If $n > m$, the function does not have a horizontal asymptote as it will approach infinity.

• Slant Asymptotes:

  • Occur when the degree of the numerator is exactly one greater than that of the denominator.

    • To find slant asymptotes, you perform polynomial long division.

• Vertical Asymptotes:

  • These occur at values of x where the function approaches infinity or negative infinity, usually at the zeros of the denominator after canceling factors.

    • For example, for a function defined as $r(x) = \frac{(x - a)(x - b)}{(x - a)}$, the vertical asymptote occurs at $x = b$.

Sequences and Growth Types

• Standard Form of an Arithmetic Sequence:
  an = ak + d(n - k)
  - Where $(k, a_k)$ represents any ordered pair in the sequence, and d represents the common difference (the consistent amount added to each term).

• Standard Form of a Geometric Sequence:
  gn = gk \cdot r^{(n-k)}
  - Where $(k, g_k)$ is any ordered pair in the sequence and r is the common ratio (the consistent factor multiplied at each term).

• Exponential Decay:

  • Occurs when $0 < b < 1$; it results in outputs that approach zero (the x-axis) as inputs increase.

• Exponential Growth:

  • Occurs when $b > 1$; it results in outputs that grow rapidly away from the x-axis as inputs increase.

Logarithmic and Trigonometric Properties

• Properties of Logarithms:

  • These properties simplify calculations involving logs:

    • \text{log}a(xy) = \text{log}a x + \text{log}_a y

      • The log of a product is the sum of the logs.

    • \text{log}a \left( \frac{x}{y} \right) = \text{log}a x - \text{log}_a y

      • The log of a quotient is the difference of the logs.

    • \text{log}a(x^n) = n \text{log}a x

      • The log of a power is the exponent times the log of the base.

• Pythagorean Trig Identity:

  • This fundamental identity expresses the relationship between sine and cosine: \text{sin}^2(x) + \text{cos}^2(x) = 1.

    • This identity is crucial for solving many trigonometric equations and simplifying expressions.

• Trigonometric Sum Formulas:

  • Fundamental formulas to combine sine and cosine functions:

    • \text{sin}(\alpha + \beta) = \text{sin}\alpha \text{cos}\beta + \text{cos}\alpha \text{sin}\beta

      • This helps in finding the sine of the sum of two angles.

    • \text{sin}(\alpha - \beta) = \text{sin}\alpha \text{cos}\beta - \text{cos}\alpha \text{sin}\beta

      • This helps in finding the sine of the difference of two angles.

    • \text{cos}(\alpha + \beta) = - \text{cos}\alpha \text{cos}\beta - \text{sin}\alpha \text{sin}\beta.

      • Useful for finding the cosine of the sum of two angles.

• Vertical Asymptotes of Trigonometric Functions:

  • Functions such as secant, cosecant, and cotangent exhibit periodic vertical asymptotes, where they approach infinity whenever their corresponding sine or cosine functions equal zero.

Polar Coordinates and Arc Length

• Conversion from Polar to Cartesian:

  • To convert points from polar coordinates (r, θ) to Cartesian coordinates (x, y):

    • $x = r \text{cos} \theta$

    • $y = r \text{sin} \theta$

    • This transformation is essential for graphing polar equations using the Cartesian system.

• Arc Length in Polar Coordinates:

  • The formula for arc length, measured in polar coordinates, is given by \theta\cdot r, which integrates the radius over a specified interval of angles to obtain the total length of the arc.

    • It measures how far along a curve you travel between two angles.

Frequency vs Period

• Period:

  • Defined as the duration required for one complete cycle of outputs in periodic functions such as sine and cosine graphs; it's the length of the interval over which the function repeats itself.

• Frequency:

  • The frequency is the reciprocal of the period, measured in cycles per unit time; it indicates how many cycles occur within a given time frame.

    • Frequency (f) can be calculated using the formula ( f = \frac{1}{T} ), where T is the period.

Function Trends

• Linear Function:

  • This function has constant first differences over equal-length inputs, which results in a straight line on the graph, defined by the equation of the form $y = mx + b$ where m is the slope, and b is the y-intercept.

• Quadratic Function:

  • Exhibited by parabolic shapes in the graph where the first difference varies but the second difference remains constant; it can be expressed as $y = ax^2 + bx + c$.

• Cubic Function:

  • This function produces an S-shaped curve where the third differences are constant; it can be expressed in the form $y = ax^3 + bx^2 + cx + d$.

• Exponential Function:

  • Characterized by outputs that increase rapidly with each input; can be defined as $y = ab^x$, where b is a constant greater than one indicating growth.

• Logarithmic Function:

  • Displays outputs that increase slowly relative to their inputs; often in the form $y = a ext{log}_b(x)$, indicating that as x increases, the output increases at a diminishing rate.