Algebra

Manipulating expressions

• Know how to manipulate polynomial expressions.
  • Adding: (x^2 + 2x + 3) + (3x^2 – 3x) = 4x^2 – x + 3(x2+2x+3)+(3x2−3x)=4x2−x+3left parenthesis, x, start superscript, 2, end superscript, plus, 2, x, plus, 3, right parenthesis, plus, left parenthesis, 3, x, start superscript, 2, end superscript, minus, 3, x, right parenthesis, equals, 4, x, start superscript, 2, end superscript, minus, x, plus, 3
  • Multiplying and Factoring: (x + 2)(3x – 5) \Leftrightarrow 3x^2 + x – 10(x+2)(3x−5)⇔3x2+x−10

• Know how to solve simple linear equations.

  • For example, $2x+3=5x-7$2, x, plus, 3, equals, 5, x, minus, 7

• Know how to solve quadratic equations, such as 2x^2 + 3x – 5 = 02x2+3x−5=02, x, start superscript, 2, end superscript, plus, 3, x, minus, 5, equals, 0

  • Completing the square
  • Quadratic formula

• Know the properties of exponents.

  • x^2 y^2 = (xy)^2x2y2=(xy)2x, start superscript, 2, end superscript, y, start superscript, 2, end superscript, equals, left parenthesis, x, y, right parenthesis, start superscript, 2, end superscript
  • (2^x)(2^y) = 2^{x + y}(2x)(2y)=2x+yleft parenthesis, 2, start superscript, x, end superscript, right parenthesis, left parenthesis, 2, start superscript, y, end superscript, right parenthesis, equals, 2, start superscript, x, plus, y, end superscript

• Know how certain expressions are secretly exponentials in disguise

  • Reciprocals: For example, \dfrac{1}{x} = x^{-1}x1=x−1start fraction, 1, divided by, x, end fraction, equals, x, start superscript, minus, 1, end superscript
  • Roots: For example, \sqrt{x} = x^{1/2}√x=x1/2square root of, x, end square root, equals, x, start superscript, 1, slash, 2, end superscript

• Know what logarithms are, as well as their properties.

  • y = 2^xy=2xy, equals, 2, start superscript, x, end superscript says the same thing as \log_2(y) = xlog2(y)=xlog, start subscript, 2, end subscript, left parenthesis, y, right parenthesis, equals, x.
  • $\log (x)+\log (y)=\log (xy)$log, left parenthesis, x, right parenthesis, plus, log, left parenthesis, y, right parenthesis, equals, log, left parenthesis, x, y, right parenthesis.
  • \log(a^x) = x\log(a)log(ax)=xlog(a)log, left parenthesis, a, start superscript, x, end superscript, right parenthesis, equals, x, log, left parenthesis, a, right parenthesis.

Functions

• Know what a function is
• Know how to represent a function with a graph.

• Know the graphs of various elementary functions.

  • Linear functions
  • Quadratic functions
  • Have at least a loose idea for what the graph of an n^{\text{th}}nthn, start superscript, t, h, end superscript degree polynomial might look like.
  • Exponentials
  • Logarithms

• Know how to manipulate functions.

  • Adding functions
  • Multiplying functions

• It’s also helpful to be familiar with function terminology

  • Domain
  • Range
  • Odd and even functions
  • Periodic
  • Zeros
  • Intercepts

Geometry

• Know how to compute the area of simple shapes.

  • Rectangles
  • Triangles
  • Trapazoids
  • Circles

• Know how to compute the volume of simple 3d shapes.

  • Cube
  • Prism
  • Sphere
• Know how to think about the cross section of a 3d shape

• Analytic geometry

  • Reasoning about points, lines, and shapes in the coordinate plane.
  • Equations for parallel and perpendicular lines.
  • Equation of a circle

Trigonometry

• Be comforatble with each of the basic trigonometry functions: $\sin (x)$sine, left parenthesis, x, right parenthesis,$\cos (x)$cosine, left parenthesis, x, right parenthesis and $\tan (x)$tangent, left parenthesis, x, right parenthesis
  • Know what each one represents.
  • Know the values of these functions when $x$x takes on one of the following values: $0$0, \dfrac{\pi}{6}6πstart fraction, pi, divided by, 6, end fraction, \dfrac{\pi}{4}4πstart fraction, pi, divided by, 4, end fraction, \dfrac{\pi}{3}3πstart fraction, pi, divided by, 3, end fraction, \dfrac{\pi}{2}2πstart fraction, pi, divided by, 2, end fraction.
  • Know what the graph of each of these functions looks like.