Algebra 1 Section 12.7

Definition A quadratic function is a function that can be written in the form f(x) = ax2 + bx + c, where a, b, and c are real numbers and a ≠ 0. This form is called the standard form of a quadratic function.

Graphing Quadratic Functions Find at least five ordered pairs that satisfy the function. Plot the ordered pairs on a Cartesian coordinate plane. Connect the ordered pairs with a smooth curve.

Example 1 Make a table for each function. x f(x) = x2 g(x) = 3x2 1 1 3 1 1 3 -1 1 3 2 4 12 -2 4 12

Example 1 This makes the graph of g(x) appear “steeper” or “narrower” than the graph of f(x). Notice that each point on g(x) is three times higher than its corresponding point on f(x).

Example 2 p(x) = -x2 and q(x) = ½x2 x p(x) = -x2 q(x) = ½x2 1 -1 ½ -1 1 -1 ½ -1 -1 ½ 2 -4 2 -2 -4 2

Parabolas The “turning point” of a parabola is called its vertex. The vertex is either the lowest (minimum) or highest (maximum) point on the graph.

Parabolas The parabolas in Examples 1 and 2 are all symmetric across the y-axis. This “fold line” is called the line of symmetry.

Effect of a on the Graph y = ax2 + bx + c If a > 0, the graph opens upward. If a < 0, the graph opens downward.

Effect of a on the Graph y = ax2 + bx + c If |a| > 1, the graph is “steeper.” If |a| < 1, the graph is “flatter.”

Example 3 Graph and compare. x f(x) = x2 r(x) = x2 + 2 -2 4 6 -1 1 3 2 2 1 1 3 2 4 6

Example 3 The graph of r(x) is the result of translating the graph of f(x) two units up. Each point on r(x) is two units above its corresponding point on f(x).

Example 4 Graph and compare. x g(x) = 3x2 s(x) = 3x2 – 4 -2 12 8 -1 3 -4 1 3 -1 2 12 8

Example 4 The graph of s(x) is the result of translating the graph of g(x) down four units. Each point on s(x) is four units below its corresponding point on g(x).

Effect of Adding a Constant y = ax2 + c If c is positive, the graph slides up c units. If c is negative, the graph slides down c units. The vertex is at (0, c).

Homework: pp. 513-515