Chapter 8: Graphing Quadratic Functions

Table of Contents

• 8.1 Graphing f(x) = ax²
• 8.2 Graphing f(x) = ax² + c
• 8.3 Graphing f(x) = ax² + bx + c
• 8.4 Graphing f(x) = a(x - h)² + k
• 8.5 Using Intercept Form
• 8.6 Comparing Linear, Exponential, and Quadratic Functions

8.1 Graphing f(x) = ax²

• Characteristics of quadratic functions with form y = ax².
• Graph is a parabola.
• Steps to graph: Make a table of values, plot, and draw the curve.
• Vertex at origin for y = x².
• Vertical shrink/stretch affects the graph's width.

8.2 Graphing f(x) = ax² + c

• Adding 'c' translates the graph vertically.
• Vertex form: (0, c).
• Steps to graph transformations.

8.3 Graphing f(x) = ax² + bx + c

• Graph opens up/down based on 'a'.
• Axis of symmetry formula: x = -b/2a.
• Steps to find the vertex and graph the parabola.
• Maximum/minimum values based on vertex.

8.4 Graphing f(x) = a(x - h)² + k

• Vertex form allows easier graphing.
• Horizontal translation by 'h', vertical by 'k'.
• Identifying even/odd functions.

8.5 Using Intercept Form

• Intercept form: f(x) = a(x - p)(x - q).
• Used to find x-intercepts easily.
• Steps to graph using intercepts and axis of symmetry.

8.6 Comparing Linear, Exponential, and Quadratic Functions

• Differences in growth rates.
• Quadratic: y = ax² + bx + c
• Linear: y = mx + b
• Exponential: y = ab^x
• Use average rates of change for comparisons.
• Differences and ratios help identify function type.

Key Concepts

• A quadratic function's graph is a parabola.
• The vertex is the highest or lowest point.
• The axis of symmetry divides the parabola into two mirrored halves.
• Intercept form is useful for finding zeros.
• Average rate of change can distinguish function types.

Problem Solving Strategies

• Graphing calculators help verify solutions.
• Try special cases and transformations to understand graphs.
• Use technology for precise graphing and analysis.

Mathematical Practices

• Interpreting graphs and functions.
• Using transformations to describe graphs.

Exercises & Examples

• Evaluating expressions, graphing equations, finding intercepts, and modeling real-life problems using quadratic functions.
• Comparing functions using visual and analytical methods.

Study Tips

• Graphing by hand strengthens understanding.
• Practice finding vertex and axis of symmetry.
• Relate transformations to visual changes in graphs.