Section 1.2: Functions & Graphs

This section introduces functions, their graphs, domain and range, and different forms of quadratic functions for graphing purposes.

Example 1: Quadratic in Standard Form

Graph \( f(x) = x^2 - 6x + 8 \) by finding its vertex and intercepts.

Step 1: Find vertex: \( x_v = -b/2a = 6/2 = 3 \), \( y_v = f(3) = 3^2 - 18 + 8 = -1 \)

Step 2: x-intercepts: Solve \( x^2 - 6x + 8 = 0 → (x-2)(x-4) = 0 → x = 2,4 \)

Step 3: y-intercept: \( f(0) = 8 \)

Example 2: Quadratic in Factored Form

Graph \( g(x) = x^2 - 5x + 6 \) using its factored form.

Step 1: Factor: \( x^2 - 5x + 6 = (x-2)(x-3) \)

Step 2: x-intercepts: x = 2, x = 3

Step 3: Vertex: Midpoint of x-intercepts: \( x_v = (2+3)/2 = 2.5 \), \( y_v = g(2.5) = -0.25 \)

Practice Problems

  1. Graph \( f(x) = x^2 + 4x + 3 \) using vertex and intercepts
  2. Factor \( h(x) = x^2 - 7x + 10 \) and graph
  3. Find the vertex and x-intercepts of \( f(x) = -2x^2 + 8x - 3 \)
  4. Graph \( f(x) = (x+2)^2 - 4 \)
  5. Determine vertex and intercepts of \( f(x) = x^2 - 9 \)
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