Objective Graph and transform quadratic functions.

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Presentation transcript:

Objective Graph and transform quadratic functions.

The quadratic parent function is f(x) = x2 The quadratic parent function is f(x) = x2. The graph of all other quadratic functions are transformations of the graph of f(x) = x2. For the parent function f(x) = x2: The axis of symmetry is x = 0, or the y-axis. The vertex is (0, 0) The function has only one zero, 0.

Example 1B: Comparing Widths of Parabolas Order the functions from narrowest graph to widest. f(x) = x2, g(x) = x2, h(x) = –2x2 Step 1 Find |a| for each function. |1| = 1 |–2| = 2 Step 2 Order the functions. h(x) = –2x2 The function with the narrowest graph has the greatest |a|. f(x) = x2 g(x) = x2

Example 1B Continued Order the functions from narrowest graph to widest. f(x) = x2, g(x) = x2, h(x) = –2x2 Check Use a graphing calculator to compare the graphs. h(x) = –2x2 has the narrowest graph and  g(x) = x2 has the widest graph.

Check It Out! Example 1b Order the functions from narrowest graph to widest. f(x) = –4x2, g(x) = 6x2, h(x) = 0.2x2 Step 1 Find |a| for each function. |–4| = 4 |6| = 6 |0.2| = 0.2 Step 2 Order the functions. g(x) = 6x2 The function with the narrowest graph has the greatest |a|. f(x) = –4x2 h(x) = 0.2x2

Check It Out! Example 1b Continued Order the functions from narrowest graph to widest. f(x) = –4x2, g(x) = 6x2, h(x) = 0.2x2 Check Use a graphing calculator to compare the graphs. g(x) = 6x2 has the narrowest graph and  h(x) = 0.2x2 has the widest graph.

Example 2A: Comparing Graphs of Quadratic Functions Compare the graph of the function with the graph of f(x) = x2. g(x) = x2 + 3 Method 1 Compare the graphs. The graph of g(x) = x2 + 3 is wider than the graph of f(x) = x2. The graph of g(x) = x2 + 3 opens downward and the graph of f(x) = x2 opens upward.

Example 2A Continued Compare the graph of the function with the graph of f(x) = x2 g(x) = x2 + 3 The vertex of f(x) = x2 is (0, 0). g(x) = x2 + 3 is translated 3 units up to (0, 3). The axis of symmetry is the same.

Check It Out! Example 2c Compare the graph of the function with the graph of f(x) = x2. g(x) = x2 + 2 Method 1 Compare the graphs. The graph of g(x) = x2 + 2 is wider than the graph of f(x) = x2. The graph of g(x) = x2 + 2 opens upward and the graph of f(x) = x2 opens upward.

Check It Out! Example 2c Continued Compare the graph of the function with the graph of f(x) = x2. g(x) = x2 + 2 The vertex of f(x) = x2 is (0, 0). g(x) = x2 + 2 is translated 2 units up to (0, 2). The axis of symmetry is the same.