Standard MM2A3. Students will analyze quadratic functions in the forms f(x) = ax2 + bx + c and f(x) = a(x – h)2 + k. c. Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase and decrease, and rates of change

The graph of a quadratic equation is a PARABOLA. MM2A3c c. Investigate and explain characteristics of quadratic functions A quadratic function is a function that can be written in standard form: y = ax2 + bx + c where a is not equal to 0. The graph of a quadratic equation is a PARABOLA.

MM2A3c c. Investigate and explain characteristics of quadratic functions y x Parent Quadratic Function: f(x) = x2 Let’s graph it with a table of values!! x f(x) Now let’s describe it!!

The lowest or highest point on a parabola MM2A3c c. Investigate and explain characteristics of quadratic functions Let’s define!! Vertex: The lowest or highest point on a parabola In our parent function example: Vertex: (0,0)

In our parent function example: MM2A3c c. Investigate and explain characteristics of quadratic functions Axis of Symmetry: An invisible vertical line that divides the parabola into mirror images and passes through the vertex. (x = ____) In our parent function example: x = 0

The set of all input (x) values of a relation MM2A3c c. Investigate and explain characteristics of quadratic functions Domain: The set of all input (x) values of a relation In our parent function example: Domain = all real numbers or

The set of all output (y) values of a relation MM2A3c c. Investigate and explain characteristics of quadratic functions Range: The set of all output (y) values of a relation In our parent function example: Range = or

Where the graph of a function crosses or touches the x-axis MM2A3c c. Investigate and explain characteristics of quadratic functions Zero(s): Where f(x) = 0; Where the graph of a function crosses or touches the x-axis In our parent function example: Zero: x = 0

The minimum(s) and maximum(s) of a function on a certain interval. MM2A3c c. Investigate and explain characteristics of quadratic functions Extrema: The minimum(s) and maximum(s) of a function on a certain interval. The vertex’s y-coordinate is the: MINIMUM value if a>0 MAXIMUM value if a<0 In our parent function example: Extrema: Minimum at y = 0

Interval(s) of Increase: From left to right on a graph, MM2A3c c. Investigate and explain characteristics of quadratic functions Interval(s) of Increase: From left to right on a graph, where as x increases, f(x) increases In our parent function example: Int. of Increase = x > 0 Or

Interval(s) of Decrease: From left to right on a graph, MM2A3c c. Investigate and explain characteristics of quadratic functions Interval(s) of Decrease: From left to right on a graph, where as x increases, f(x) decreases In our parent function example: Int. of Increase = x < 0 Or

Point(s) where the function crosses or touches the x-axis MM2A3c c. Investigate and explain characteristics of quadratic functions X-intercept(s): Point(s) where the function crosses or touches the x-axis In our parent function example: X-intercept: x = 0

Point(s) where the function crosses or touches the y-axis MM2A3c c. Investigate and explain characteristics of quadratic functions Y-intercept(s): Point(s) where the function crosses or touches the y-axis In our parent function example: y-intercept: y = 0

MM2A3c c. Investigate and explain characteristics of quadratic functions Average Rate of Change: The slope of the line that passes through two given points on the function In our parent function example: On the interval: * Point 1 = (0,0) * Point 2 = (2,4) The average rate of change is 2

Characteristics of Quadratic Functions Homework: Characteristics of Quadratic Functions Practice Sheet WRITE PROBLEMS and SHOW WORK for credit!!