Big Ideas Math: Chapter 2 Flashcards

Study Notes

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.

Description

Explore key concepts from Chapter 2 of Big Ideas Math with these flashcards. Learn about standard form of parabolas and their horizontal and vertical translations. This quiz is designed to help you understand quadratic functions and their properties in a visually engaging way.