Big Ideas Math: Chapter 2 Flashcards

Standard Form

• Standard form of a quadratic equation is expressed as y = ax² + bx + c.
• The coefficient c represents the y-intercept of the parabola.

Horizontal Translations

• The base function f(x) = x² can be translated horizontally using f(x-h) = (x-h)².
• A negative h value (h < 0) moves the graph left, while a positive h value (h > 0) shifts it right.

Vertical Translations

• Vertical translation is described by f(x) + k = x² + k.
• The graph shifts upward when k > 0 and downward when k < 0.

Reflections

• Reflection in the x-axis can be expressed as -f(x) = -x², which flips the parabola over the x-axis.
• Reflection in the y-axis, f(-x) = x², does not change the shape unless the vertex is altered by h and k.

Horizontal Stretches and Shrinks

• A function can be altered horizontally with f(ax) = (ax)²; a value 0 < a < 1 results in a stretch, while a > 1 leads to a shrink.

Vertical Stretches and Shrinks

• Vertical adjustments are represented by a(f(x)) = ax²; where a > 1 creates a stretch and 0 < a < 1 leads to a shrink.

Vertex Form

• The vertex form of a parabola is y = a(x-h)² + k.
• Here, (h, k) denotes the vertex, with a indicating potential reflections and scaling.

Axis of Symmetry

• The axis of symmetry for a parabola is a vertical line that intersects the vertex at x = h.
• It can also be derived using x = -b/(2a) from the standard form y = ax² + bx + c.

Intercept Form

• Expressed as y = a(x-p)(x-q), where p and q are the x-intercepts of the graph.
• The axis of symmetry is derived as x = (p+q)/2, while the parabola opens upwards if a > 0 and downwards if a < 0.

Minimum and Maximum Values

• The y-coordinate of the vertex indicates the minimum value for a > 0 and maximum for a < 0.
• The domain of both cases includes all real numbers, while the range is defined as y ≥ h or y ≤ h.

Parabola Definition

• A parabola consists of points equidistant from a focus and a directrix.
• The vertex lies midway between the focus and the directrix, with the focus located on the axis of symmetry.

Standard Equations of Parabola

• For a vertical axis of symmetry at the origin, the equation is y = 1/(4p)x², with the focus at (0, p) and directrix y = -p.
• For a horizontal axis of symmetry at the origin, the equation is x = 1/(4p)y², with the focus at (p, 0) and directrix x = -p.

Parabola with Vertex at (h, k)

• Vertical axis of symmetry: y = 1/(4p)(x-h)² + k; focus at (h, k+p) and directrix y = k-p.
• Horizontal axis of symmetry: x = 1/(4p)(y-k)² + h; focus at (h+p, k) and directrix y = h-p.

Data Characteristics

• Linear data exhibit constant first differences, whereas quadratic data display constant second differences.

Writing Quadratic Equations

• Use vertex form y = a(x-h)² + k for a point given the vertex (h, k).
• Use intercept form y = a(x-p)(x-q) for a point given x-intercepts p and q.
• If three points are provided, create and solve a system of three equations.