Analyzing Graphs of Quadratic Functions

This set of flashcards covers key vocabulary and concepts related to analyzing and graphing quadratic functions.

Vertex

The point at which the graph of a quadratic function turns.

Axis of Symmetry

A line that divides the parabola into two mirror images.

Completing the Square

A method used to find the vertex and maximum or minimum values of a quadratic function.

Parabola

The graph of a quadratic function.

Maximum value

The highest point of the graph, occurring at the vertex for a downward-opening parabola.

Minimum value

The lowest point of the graph, occurring at the vertex for an upward-opening parabola.

Quadratic Function

A function in the form of f(x) = ax^2 + bx + c.

Range

The set of possible output values (y-values) of a function.

Increasing interval

The interval where the function values are rising.

Decreasing interval

The interval where the function values are falling.

Maximizing Area

The process of finding the maximum area in applications involving quadratic functions.

Area of a Rectangle

Calculated as length times width.

Steps to Solve Quadratic Function Problems

1. Understand the Problem: Identify what the question is asking (e.g., find vertex, max/min value, intercepts, range, maximize area). 2. Formulate the Quadratic Function: If not provided, express the problem as a quadratic function in the form f(x) = ax^2 + bx + c. 3. Find the Vertex: Use the formula x = -b/(2a) to find the x-coordinate of the vertex. Substitute this x-value into the function to find the y-coordinate. 4. Identify the Axis of Symmetry: This is the vertical line x = -b/(2a). 5. Determine Maximum or Minimum Value: - If a > 0 (parabola opens upwards), the y-coordinate of the vertex is the minimum value. - If a < 0 (parabola opens downwards), the y-coordinate of the vertex is the maximum value. 6. Find Intercepts (if necessary): - To find x-intercepts, set f(x) = 0 and solve for x. - To find the y-intercept, set x = 0 and calculate f(0). 7. Determine Range and Increasing/Decreasing Intervals: - Range: For a > 0, range is $$[y_{vertex}, \