Pre-Calc

Formula Sheet that we will get for the final:

    Sum and Difference

    Half-Angle and Double Angle

    Pythagorean Identities

    Law of Sins and Cosines

    Area of Oblique Triangles

    Sequences and Nth Partial Sums

    Derivatives

Formulas to memorize:

• Equation of a line tangent to a curve of the function F

  • y=f’(a)(x-a) + f(a)

  • x=a

• Transformations

  • Shift up: )+2 Shift down: )-2

  • Shift right: (x-4) Shift left: (x+4)

  • Vertical Stretch: 4(x) Vertical Shrink: ¼ (x)

  • Horizontal Stretch: 4x Horizontal Shrink: ¼ x

  • Reflect over y-axis: -x Reflect over x-axis: -(x)

• Logarithmic Form

  • \log_{b}x

• Exponential Form

  • b^{y} =x

• Imaginary Number (I won, I won)- (i -1, -i 1)

  • i= \sqrt{_{-1}} or just i

  • i² = -1

  • i³ = -i

  • i ^4 = 1

• Relation Definitions: a set of ordered pairs

• The first set of components is the domain (x)

• The second set of components in the range (y)

• Need to pass the vertical line test

  • Cannot go through the same function 2 times or more vertically

• Local Maximum:

  • graph of a function is increasing to the left of x=c and decreasing to the ight of x=c

  • Value is at the max

• Local Minimum:

  • graph of a function is decreasing to the left of x=c and increasing to the right of x=c

  • Value is at the minimum

• Sequences:

  • Arithmetic

    • Recursive- An= An-1 +d

    • D is the common difference

    • Explicit- An=An-1 +d(n-1)

  • Geometric

    • Recursive- An=An-1\cdot r

    • R is the common ratio

    • Explicit- An= A1\cdot r^{n-1}

• Series

  • Arithmetic

    • Sn= \frac{n\left(a1+an\right)}{2}

  • Geometric

    • Sn= \frac{a1\left(1-r^{n}\right)}{1-r}

• Inverse Functions

  • Set the equation to x and find y.

    • One-to-one functions     

      • If it passes the vertical line test, then its a function

    • If a function passes the HLT as well then it is considered one-to-one

• Inverse Trig Functions

  • Quadrants

    • Sine: QI and QIV

    • Cosine: QI and QII

    • Tangent: QI and QIV

• Trig Equations

  • Remember the “Let u= cos, sin, or tan (x)”

  • \sin^2 +\cos^2 +1

    • \sin^2\theta = 1-\cos^2\theta

      The same thing if you flip cos and sin

• Derivatives- a way to represent rate of change by which a function is changing at one given point.

  • Finding Limits- \lim_{x\to c}f\left(x\right)=L

• Angles are always measured from the initial side to the terminal side in a counter-clockwise path

  • Degrees to radians:

    • \frac{\pi}{180}\cdot degrees

  • Radians to degrees:

    • \frac{180}{\pi}\cdot radians

• Rational Zero Theorem

  • Possible Rational Zeros= \frac{p}{q}, or Factors of the constant term over factors of the leading coefficient

• Descartes Rule of Signs

  • Counting the number of sign changes in the sequence

  • Subtract 2 each times to find other possibilities

• Steps of sketching Polynomials Practice

  • 1: End Behavior

  • 2: Y-intercept

  • 3: Descartes’ Rule of Sign

  • 4: Rational Zero Theorem

  • 5: Factor Theorem

  • 6: Quadratic Formula

  • 7:Sketch

• Reciprocal Function

  • Local Behavior:

    • As → 0^{+}

    • As x approaches 0 from the right F(x) → \infty

    • As x→ 0^{-}

    • As x approaches 0 from the left f(x) → -\infty

  • End Behavior

    • As x→ \infty

      • f(x)→ 0

    • As x→ -\infty

      • f(x)→ 0