Algebra Midterm #1 Review

• §7.1: Introduction to Functions
  • Definitions (relation, function, domain, range)
  • Function notation: f(input) = output
  • Function as a graph, as an equation, as a data table


• §7.2: Domain and Range
  • Finding domain and range from graph or from data table
  • Finding domain from equation based on inputs that "don't work"
  • Using interval notation [a,b] or (a,b) to describe domain and range


• §7.3: Functions and Graphing
  • The graph represents the set of all points that "fit" in the function
  • In other words, all (x,y) values that make the equation true
  • Generally: choose x-values as inputs, calculate outputs as y-values, plot points
  • Standard form (Ax+By=C) and slope-intercept form (y=mx+b) for linear functions


• §7.4: Function Arithmetic
  • (f+g)(x), (f–g)(x), (f·g)(x), (f/g)(x)
  • Each is shorthand for f(x)+g(x), f(x)–g(x), etc.
  • Division may introduce new restrictions on domain


• §7.5: Formulas and Variation
  • Direct variation: output = constant · input
  • Inverse variation: output = constant / input
  • Join variation: output = constant · input₁ · input₂
  • Solving a formula for a variable: isolate that variable on one side of the equation
  • If it occurs on both sides of the equation, get all occurrences to one side first
  • If necessary, multiply by a common denominator to clear fractions

• §8.1: 2x2 Systems of Equations
  • Meaning of "solution" for a system of equations
  • What the solution means on a graph, and how to find it by graphing


• §8.2: Substitution and Elimination
  • Substitution: solve one equation for a variable; substitute into other equation
  • Elimination: combine two equations to eliminate a variable, then solve for the other
  • In both cases, finish by substituting the value of the known variable back into the original equation and solving for the final unknown variable


• §8.3: Applications of 2x2 Systems
  • Mixture Problems: one equation for total amount, one for total price/concentration/etc.
  • Motion Problems: speed = distance/time; set up equation for each object


• §8.4: 3x3 Systems of Equations
  • Elimination works best here: combine equations to eliminate one variable twice
  • This produces a 2x2 system; solve it by any method
  • Now back-substitute into previous equations to find other variables
  • Be careful about inconsistent systems and dependent systems


• §8.5: Applications of 3x3 Systems


• §8.6: Augmented Matrices
  • Set up an augmented matrix to represent the system of equations
  • Coefficients in a square matrix on the left; constants in a column on the right
  • Use row operations to transform the left side into the Identity Matrix
  • The right column will become the values of the variables


• Matrix Algebra
  • Know how to multiply two matrices together
  • Know how to find the inverse of a matrix, and what that means
  • A system of equations can be written as A·X = B, where...
    • ...A is the square matrix of coefficients
    • ...X is the single-column matrix of variables
    • ...B is the single-column matrix of constants
  • Left-multiply by A^-1, and X = A^-1·B
  • The only difficulty is finding A^-1

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