Math Exam Notes

Exponential Models

• Exponential models graph and model exponential growth and decay.

• Exponential Function: y = a(b)^x

  • where 'a' is constant

  • b > 1 (growth) or 0 < b < 1 (decay)

• Growth Factor (b): b > 1

  • Quantity increases by a constant percentage each time period.

  • Percentage increase, r, is the growth rate.

  • A(t) = a(1 + r)^t

• Decay Factor (b): 0 < b < 1

  • Quantity decreases by a constant percentage each time period.

  • Percentage decrease, r, is the decay rate.

  • A(t) = a(1 - r)^t

Exponential Functions and Transformations

• Parent function: y = b^x

• Vertical stretch/compression/reflection: y = a(b)^x

  • Vertical Stretch: |a| > 1

  • Vertical Compression: 0 < |a| < 1

  • Reflection over x-axis: a < 0

• Horizontal/Vertical Translations: y = a(b)^{x-h} + k

  • h: horizontal translation

  • k: vertical translation

Logarithmic Functions

• Definition: log_b(x) = y is equivalent to b^y = x (where b > 0 and b \neq 1).

• Inverse of Exponential Function: If y = log_b(x), then x = b^y

Properties of Logarithms

• Product Property: logb(mn) = logb(m) + log_b(n)

• Quotient Property: logb(\frac{m}{n}) = logb(m) - log_b(n)

• Power Property: logb(m^n) = n \cdot logb(m)

Parabolas

• Definition: Set of all points equidistant to the focus and directrix.

• Vertex: Midpoint between focus and directrix.

• General Equations:

  • Opens right: y^2 = 4px, p > 0

  • Opens left: y^2 = 4px, p < 0

  • Opens up: x^2 = 4py, p > 0

  • Opens down: x^2 = 4py, p < 0

• Translated Parabolas (vertex at (h, k)

  • Horizontal axis: (y - k)^2 = 4p(x - h)

  • Vertical axis: (x - h)^2 = 4p(y - k)

Conic Sections

• Conic sections are curves formed by the intersection of a plane and a double-napped cone (parabola, circle, ellipse, hyperbola).

• General Equation: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0

• Classifying Conics Using the Discriminant (B^2 - 4AC):

  • Circle: B^2 - 4AC < 0, B = 0, and A = C

  • Ellipse: B^2 - 4AC < 0 and either B \neq 0 or A \neq C

  • Parabola: B^2 - 4AC = 0

  • Hyperbola: B^2 - 4AC > 0

Hyperbolas

• Set of all points such that the difference of the distances between a point P and two fixed points (foci) is constant.

• Transverse Axis: Line segment joining the vertices.

• Conjugate Axis: Line segment perpendicular to the transverse axis, passing through the center.

• Standard Form Equations:

  • Horizontal transverse axis: \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1

  • Vertical transverse axis: \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1

• Relationship: c^2 = a^2 + b^2 (where c is the distance from center to focus).

Ellipses

• Set of all points such that the sum of the distances between a point (x, y) and two fixed points (foci) is constant.

  • Major Axis: The line segment through with the foci as endpoints

  • Minor Axis: The line segment through the center, perpendicular to th major axis, with endpoints on the ellipse

• Standard Form Equations (center at (h, k)

  • Horizontal major axis: \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1

  • Vertical major axis: \frac{(x - h)^2}{b^2} + \frac{(y - k)^2}{a^2} = 1
    a>b

• Relationship: c^2 = a^2 - b^2 (where c is the distance from center to focus).

Circles

• Set of all points equidistant from a center point.

• Standard Form: (x - h)^2 + (y - k)^2 = r^2 (center (h, k), radius r).

Standard Deviation

• Variance: sx^2 = \frac{\sum(xi - \bar{x})^2}{n-1}

• Standard Deviation: sx = \sqrt{\frac{\sum(xi - \bar{x})^2}{n-1}}

Normal Distribution and the 68-95-99.7 Rule

• Approximately 68% of data falls within 1 standard deviation of the mean.

• Approximately 95% of data falls within 2 standard deviations of the mean.

• Approximately 99.7% of data falls within 3 standard deviations of the mean.

Analyzing Data

• Measures of Central Tendency:

  • Mean: Average of the data (\bar{x} = \frac{\sum x_i}{n})

  • Median: Middle data value.

  • Mode: Most frequent data value(s).

• Five-Number Summary: Minimum, Q1, Median, Q3, Maximum.

• Outliers:

  • Interquartile Range (IQR): Q3 - Q1

  • Lower Boundary: Q1 - 1.5 \cdot IQR

  • Upper Boundary: Q3 + 1.5 \cdot IQR

• Percentile Rank: PR = \frac{b}{n} \times 100 (b = number of values below, n = total number of values).

Z-score

• z = \frac{x - \mu}{\sigma}