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Precalculus Key Concepts and Techniques

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  • What is the effect of transformations on the graph of a function?
    Transformations shift, stretch, compress, or reflect the graph of a function to produce a new graph.
  • How do you simplify complex numbers?
    Combine like terms and use \(i^2 = -1\) to rewrite powers of i.
  • What is a radical equation?
    An equation in which the variable is inside a root, such as a square root.
  • How do you solve rational equations?
    Find a common denominator, multiply both sides to clear fractions, then solve the resulting equation.
  • What is an absolute value equation?
    An equation involving the absolute value function, which measures distance from zero.
  • How do you solve quadratic equations using multiple techniques?
    Use factoring, completing the square, quadratic formula, or graphing.
  • How do you find the vertex of a quadratic function?
    Use the formula \(x = -\frac{b}{2a}\) to find the x-coordinate, then substitute to find y.
  • What are the properties of logarithms used to solve equations?
    Properties include product, quotient, and power rules to simplify and solve logarithmic equations.
  • How do you solve polynomial equations?
    Factor the polynomial or use the Rational Root Theorem and synthetic division.
  • What is the general shape of exponential functions?
    Exponential functions have a curve that increases or decreases rapidly, depending on the base.
  • How do you find zeros of rational functions?
    Set the numerator equal to zero and solve for the variable.
  • What are vertical asymptotes in rational functions?
    Vertical lines where the function is undefined, found by setting the denominator equal to zero.
  • What is a composite function?
    A function formed by applying one function to the result of another, denoted as \((f \circ g)(x) = f(g(x))\).
  • How do you find the inverse of a function?
    Swap the roles of x and y in the equation and solve for y.
  • What is function notation?
    A way to denote functions using \(f(x)\) to represent the output for input x.
  • How do you determine the domain of a function?
    Find all input values for which the function is defined.
  • How do you determine the range of a function?
    Find all possible output values of the function.
  • What does it mean for a function to be increasing or decreasing?
    Increasing means output values rise as input increases; decreasing means output values fall.
  • How do you find the maximum or minimum of a function?
    Identify the highest or lowest point on the graph, often the vertex for quadratics.
  • What is the difference quotient?
    A formula \(\frac{f(x+h)-f(x)}{h}\) used to find the average rate of change of a function.
  • What types of equations are quadratic in form?
    Equations that can be rewritten to resemble a quadratic equation, often by substitution.
  • How do you solve absolute value inequalities?
    Split into two inequalities and solve each separately.
  • What is the role of technology in precalculus?
    Technology helps graph functions, solve equations, and visualize concepts.
  • What are rational expressions?
    Expressions that are ratios of polynomials.
  • How do you simplify rational expressions?
    Factor numerator and denominator and cancel common factors.
  • What is the significance of vertical asymptotes in graphing rational functions?
    They indicate values where the function grows without bound or is undefined.
  • How do you sketch polynomial functions?
    Identify zeros, end behavior, and turning points to draw the graph.
  • What is the domain restriction for radical functions?
    The radicand must be greater than or equal to zero for even roots.