Quadratic Functions and Their Graphs: Section 8.6 Study Notes

Quadratic Functions and Their Graphs

Introduction to Quadratic Functions

Quadratic functions are fundamental in algebra and are characterized by their parabolic graphs. The general form of a quadratic function is f(x) = ax^2, where a is a real number. Understanding how to graph these functions and their transformations is essential for Intermediate Algebra students.

Graphing Basic Quadratic Functions

The graph of f(x) = ax^2 is a parabola. The properties of the graph depend on the value of a:

• Vertex: The turning point of the parabola, located at (0, 0) for f(x) = ax^2.

• Axis of Symmetry: The vertical line x = 0 divides the parabola into two symmetric halves.

• Direction: The parabola opens upward if a > 0 and downward if a < 0.

• Width: The parabola is narrower if |a| > 1 and wider if 0 < |a| < 1.

Example: Graph f(x) = x^2 and label the vertex and axis of symmetry.

Transformations of Quadratic Functions

Quadratic functions can be transformed by shifting, stretching, or compressing their graphs. The general form f(x) = a(x-h)^2 + k allows for these transformations:

• Horizontal Shift: If h is positive, the graph shifts right; if h is negative, it shifts left.

• Vertical Shift: If k is positive, the graph shifts up; if k is negative, it shifts down.

• Vertex: The vertex is at (h, k).

• Axis of Symmetry: The line x = h is the axis of symmetry.

• Minimum/Maximum Value: If the parabola opens up, k is the minimum value; if it opens down, k is the maximum value.

Example: Graph f(x) = 2(x-3)^2 + 1. State the vertex, axis of symmetry, and minimum/maximum value.

Comparing Quadratic Function Transformations

Different forms of quadratic functions result in different transformations. Consider the following:

• f(x) = ax^2: Basic parabola centered at the origin.

• g(x) = ax^2 + 1: Vertical shift upward by 1 unit.

• p(x) = a(x+1)^2: Horizontal shift left by 1 unit.

• h(x) = -ax^2: Reflection across the x-axis.

• q(x) = 2ax^2: Vertical stretch, making the parabola narrower.

Example: Match each transformation to its graph.

Summary Table: Properties of Quadratic Function Graphs

Function Form	Vertex	Axis of Symmetry	Direction	Width	Shift
$f(x) = ax^2$	(0, 0)	x = 0	Up if $a > 0$, Down if $a < 0$	Narrow if $|a| > 1$, Wide if $0 < |a| < 1$	None
$f(x) = a(x-h)^2$	(h, 0)	x = h	Up if $a > 0$, Down if $a < 0$	Narrow if $|a| > 1$, Wide if $0 < |a| < 1$	Horizontal
$f(x) = a(x-h)^2 + k$	(h, k)	x = h	Up if $a > 0$, Down if $a < 0$	Narrow if $|a| > 1$, Wide if $0 < |a| < 1$	Horizontal & Vertical

Practice Problems

• Graph f(x) = 4x^2. State the vertex and axis of symmetry.

• Graph f(x) = -4x^2. State the vertex and axis of symmetry.

• Explain how to obtain the graph of g(x) = (x-1)^2 from f(x) = x^2.

• Explain how to obtain the graph of p(x) = x^2 + 10 from f(x) = x^2.

• Explain how to obtain the graph of h(x) = 8(x+2)^2 - 3 from f(x) = x^2.

• For f(x) = (x+3)^2 - 1, find the vertex, axis of symmetry, and minimum/maximum value, then graph the function.

Key Points and Common Mistakes

• Always identify the vertex and axis of symmetry before graphing.

• Check the sign of a to determine the direction the parabola opens.

• Remember that horizontal shifts are determined by h and vertical shifts by k.

• Plot several points to ensure accuracy when sketching the graph.