Section 2.5: Review & Assessment
This section reviews all key concepts from Chapter 2: Quadratic Functions. Complete the examples and practice problems to assess your understanding of graphing, vertex form, quadratic formula, and real-world applications.
Example 1: Graphing a Quadratic Function
Graph \( f(x) = x^2 - 6x + 8 \) and identify the vertex, axis of symmetry, and x-intercepts.
Step 1: Rewrite in vertex form: \( f(x) = (x-3)^2 -1 \)
Step 2: Vertex: (3, -1)
Step 3: Axis of symmetry: x = 3
Step 4: Factor to find x-intercepts: \( x^2 -6x +8 = (x-2)(x-4) \) → x = 2, 4
Example 2: Using Quadratic Formula
Solve \( 2x^2 - 4x - 6 = 0 \) using the quadratic formula.
Step 1: Identify coefficients: a=2, b=-4, c=-6
Step 2: Apply formula: \( x = [-(-4) ± \sqrt{(-4)^2 -4(2)(-6)}]/(2*2) \)
Step 3: Simplify: \( x = [4 ± \sqrt{16+48}]/4 = [4 ± \sqrt{64}]/4 \)
Step 4: Final answer: \( x = (4±8)/4 → x=3, x=-1 \)
Example 3: Real-World Application
A ball is thrown with height \( h(t) = -16t^2 + 48t + 80 \). Find the maximum height and the time it occurs.
Step 1: Vertex formula: t = -b/(2a) = -48/(2*-16) = 1.5 seconds
Step 2: Maximum height: h(1.5) = -16(1.5)^2 + 48(1.5) + 80 = 116 ft
Practice Problems
- Graph \( f(x) = x^2 + 4x -5 \) and find vertex and x-intercepts
- Rewrite \( g(x) = 3x^2 -12x +7 \) in vertex form
- Solve \( x^2 + 6x + 5 = 0 \) by factoring
- Find the maximum height of \( h(t) = -9.8t^2 + 19.6t +10 \)
- Determine axis of symmetry and end behavior for \( f(x) = -2x^2 + 8x -3 \)