Unit 10 – Quadratic Functions Topic: Characteristics of Quadratic Functions

What is a quadratic function?  Standard form:  Parent quadratic function:  Graph: parabola

What is the vertex of a quadratic function?  Highest or lowest point  Vertex: (-1, -6) o y-value is called minimum o Parabola opens upward (a > 0)

What is the vertex of a quadratic function?  Vertex: (1, 7) o y-value is called maximum o Parabola opens downward (a < 0)

Finding domain & range  Domain: ALWAYS all real #  Range: ALWAYS an inequality –y coordinate of vertex represents minimum or maximum value of range  Range: y ≥ -6

Finding domain & range  Domain: all real #  Range: y ≤ 7

What is the axis of symmetry?  Vertical line that divides parabola in half –REMEMBER: equation for a vertical line is x = a  a of s: x = -1

Finding axis of symmetry algebraically  Formula:  Example: Find the axis of symmetry for the function Plug in values for a (2) & b (–8) & simplify. WATCH YOUR SIGNS! Axis of symmetry for this function is the vertical line x = 2. SIGN NOTE: Notice the two negatives cancel. Remember the formula includes a negative.

Using axis of symmetry to find vertex  Finding vertex coordinates: –x-coordinate: axis of symmetry –y-coordinate: substitute x-coordinate into function & simplify We’ve already found the x-coordinate (2). Replace x in the function with 2 & solve for y. Vertex for this function is the point (2, –11).

What are the zeros of a quadratic function?  x-value(s) that makes function = 0  Using graph: zeros are the points where the parabola crosses x-axis o Two real zeros o x = -1 and x = 2

What are the zeros of a quadratic function?  one real zero o x = 1

What are the zeros of a quadratic function?  No real zeros

Determining a Function From a Graph  Identify 3 points from the graph. –One should be the y- intercept; pick points that make the math easy.  (0, 6), (2, 0), (3, 0)

Determining a Function From a Graph  Using standard form of a quadratic equation, write a system of equations. –REMEMBER: We already have a value for c (from y-intercept).

Determining a Function From a Graph  Simplify & solve for a & b. Divide 1 st equation by -2. Divide 2 nd equation by 3. Add equations to eliminate b. Plug the value of a into one of the equations & solve for b.

Determining a Function From a Graph Write the function in standard form with the values of a, b & c. Check your equation on your graphing calculator.

Homework Complete the handout you received in class. Be prepared to present solutions on the board. DUE 4/16 (A-day) or 4/17 (B-day)