Algebra 1 Section 12 7 Definition A quadratic

Algebra 1 Section 12. 7

Definition A quadratic function is a function that can be written in the form f(x) = ax 2 + 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 1. Find at least five ordered pairs that satisfy the function. 2. Plot the ordered pairs on a Cartesian coordinate plane. 3. Connect the ordered pairs with a smooth curve.

Example 1 Make a table for each function. x f(x) = x 2 g(x) = 3 x 2 0 1 -1 2 -2 0 1 1 4 4 0 3 3 12 12

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

Example 2 p(x) = -x 2 and q(x) = ½x 2 x p(x) = -x 2 q(x) = ½x 2 0 1 -1 2 -2 0 -1 -1 -4 -4 0 ½ ½ 2 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 = ax 2 + bx + c If a > 0, the graph opens upward. If a < 0, the graph opens downward.

Effect of a on the Graph y = ax 2 + bx + c If |a| > 1, the graph is “steeper. ” If |a| < 1, the graph is “flatter. ”

Example 3 Graph and compare. x f(x) = x 2 r(x) = x 2 + 2 -2 -1 0 1 2 4 1 0 1 4 6 3 2 3 6

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

Example 4 Graph and compare. x g(x) = 3 x 2 s(x) = 3 x 2 – 4 -2 -1 0 1 2 12 3 0 3 12 8 -1 -4 -1 8

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

Effect of Adding a Constant y = ax 2 + 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