5.2.1: Resources and Key Concepts

Resources

Videos

• Completing the square
  • Intermediate Algebra - Completing the Square (Part A)
  • Intermediate Algebra - Completing the Square (Part B)
• Graphing quadratic functions in standard/vertex form
  • Intermediate Algebra - Everything About Graphing Parabolas
  • Intermediate Algebra - Graphing Parabolas in Vertex Form
• Graphing quadratic functions in general form
  • Quadratic Functions - Understanding How Graphs of Parabolas are Related to Their Quadratic Functions
  • Intermediate Algebra - Graphing Quadratic Equations (Part A)
  • Intermediate Algebra - Graphing Quadratic Equations (Part B)
• Solving quadratic equations
  • Intermediate Algebra - Introduction to Quadratic Equations
  • Algebra in Review: Equations - Quadratic (Zero Factor Principle)
  • Intermediate Algebra - Solving Quadratic Equations by Factoring (Part A)
  • Intermediate Algebra - Solving Quadratic Equations by Factoring (Part B)
  • Intermediate Algebra - Quadratic Formula (Part A)
  • Intermediate Algebra - Quadratic Formula (Part B)
  • Intermediate Algebra - Quadratic Formula (Part C)
  • Intermediate Algebra - Quadratic Formula (Part D)
• Solving equations that are quadratic-in-form
  • Equations - Overview - A Review for College Algebra: Solving Equations That Are Quadratic-in-Form
• Solving quadratic inequalities
  • Inequalities - Nonlinear - A Review for College Algebra: Solving Quadratic Inequalities
  • Intermediate Algebra - Quadratic Inequalities (Part A)
  • Intermediate Algebra - Quadratic Inequalities (Part B)
  • Intermediate Algebra - Quadratic Inequalities (Part C)
  • Intermediate Algebra - Quadratic Inequalities (Part D)
• Applications involving quadratic functions
  • Quadratic Functions - Determining the Maximum and Minimum Values of Quadratic Functions
  • Intermediate Algebra - Quadratic Applications
  • Optimization Problems - Maximizing Revenue the Non-Calculus Way
• CAPSTONE!!!
  • Algebra in Review: Capstone - Quadratics

Key Concepts

Definitions

• Quadratic Function: A function of degree two, which can be written in the general form \(f(x) = ax^2 + bx + c\) where \(a, b, c\) are real numbers and \(a \neq 0\).
• Parabola: The U-shaped curve that is the graph of a quadratic function.
• Vertex (of a parabola): The point on the parabola where the function reaches its maximum or minimum value. It is also the turning point of the graph.
• Axis of Symmetry (of a parabola): A vertical line passing through the vertex of a parabola, dividing it into two mirror images. Its equation is \(x=h\), where \(h\) is the \( x \)-coordinate of the vertex.
• General Form (of a quadratic function): \(f(x) = ax^2 + bx + c\).
• Standard Form (or Vertex Form of a quadratic function): \(f(x) = a(x-h)^2 + k\), where \((h,k)\) is the vertex of the parabola.
• Zeros (or Roots or x-intercepts of a quadratic function): The values of \(x\) for which \(f(x)=0\). These are the points where the parabola intersects the \( x \)-axis.
• \( y \)-intercept (of a quadratic function): The point where the parabola intersects the y-axis, found by evaluating \(f(0)\). In general form, this is \((0,c)\).
• Discriminant: For a quadratic equation \(ax^2+bx+c=0\) (or function \(f(x)=ax^2+bx+c\)), the discriminant is the value \(D = b^2-4ac\). It helps determine the nature and number of real roots (\( x \)-intercepts).
• Prime Quadratic: A quadratic expression that cannot be factored into linear factors with real coefficients. This occurs when its discriminant is negative.
• \( u \)-substitution: A technique used to solve equations that are "quadratic-in-form" by substituting a variable (often \(u\)) for an expression, transforming the equation into a standard quadratic form.

Theorems

• Axis of Symmetry and Vertex (from General Form): For a quadratic function \(f(x) = ax^2 + bx + c\), the axis of symmetry is the line \(x = -\frac{b}{2a}\). The vertex is at \((-\frac{b}{2a}, f(-\frac{b}{2a}))\).
• Vertex (from Standard Form): For a quadratic function \(f(x) = a(x-h)^2 + k\), the vertex is \((h,k)\).
• Relationship between General and Standard Forms:
  • Given \(f(x) = ax^2 + bx + c\), then \(h = -\frac{b}{2a}\) and \(k = c - ah^2 = c - a(-\frac{b}{2a})^2 = c - \frac{b^2}{4a}\). More simply, \(k=f(h)\).
• Domain and Range of a Quadratic Function:
  • The domain of any quadratic function is all real numbers, \((-\infty, \infty)\).
  • If \(a > 0\), the range is \([k, \infty)\), where \(k\) is the y-coordinate of the vertex.
  • If \(a < 0\), the range is \((-\infty, k]\), where \(k\) is the y-coordinate of the vertex.
• Zeros and the Discriminant: For a quadratic function \(f(x) = ax^2 + bx + c\), let \(D = b^2-4ac\) be the discriminant.
  • If \(D > 0\), the function has two distinct real zeros (two x-intercepts).
  • If \(D = 0\), the function has one real zero (one x-intercept; the vertex is on the x-axis).
  • If \(D < 0\), the function has no real zeros (no x-intercepts); the zeros are complex conjugates.
• Quadratic Formula: The solutions to the quadratic equation \(ax^2+bx+c=0\) (where \(a \neq 0\)) are given by \(x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\).

Common Mistakes to Avoid

• Confusing \(h\) in Standard Form: In \(f(x)=a(x-h)^2+k\), if the form is, for example, \(f(x)=a(x+3)^2+k\), misidentifying \(h\) as \(3\) instead of \(-3\).
• Errors in Completing the Square: Forgetting to factor out the leading coefficient \(a\) from the \(x^2\) and \(x\) terms before completing the square.
• Factoring Errors: Incorrectly factoring a quadratic trinomial when trying to find \( x \)-intercepts or solve inequalities.
• Solving Quadratic Inequalities: Forgetting to flip the inequality sign if multiplying/dividing the entire inequality by a negative number.
• Equations Quadratic-in-Form: After solving for the substituted variable (e.g., \(u\)), forgetting to substitute back to solve for the original variable (e.g., \(x\)).
• Absolute Value Equations with Quadratics: Not considering both positive and negative cases for the expression inside the absolute value (e.g., for \(|Q(x)|=c\), solving only \(Q(x)=c\)).
• Graphing \(|Q(x)|\): Reflecting the entire graph of \(Q(x)\) instead of only the parts below the \( x \)-axis.