Characteristics of Quadratic Functions CA 21.0, 23.0

Objectives - 1) To find the roots (zeros) of a quadratic function by looking at the graph. 2) To find the axis of symmetry by using the vertex formula.

The ROOTS The ROOTS of a quadratic equation are also called:  ZEROS  SOLUTIONS  x-intercepts Roots, zeros, solutions, x-intercepts ALL mean exactly the same thing!

Finding the Roots from Graphs To find the roots on a graph, look for the x-intercepts x y

x y Vertex (1,9) x = 1 x-intercepts Find the roots of The roots are x = –2 and x = 4

x y Find the x-intercepts of the equation below.

x y Find x-intercepts of Solution: x = -3 and x = 1

Finding the Axis of Symmetry If you know the roots, you can find the axis of symmetry by finding the AVERAGE of the X-VALUES Example: x y The roots are -3 and +4 Average:

What if there is only one root? x y What is the equation of the vertical line passing through the vertex?

Finding the Axis of Symmetry But what if we don’t know the roots? In order to find the AXIS OF SYMMETRY, we need to find the x-coordinate of the VERTEX. We do this by using the first part of THE VERTEX FORMULA

The AXIS OF SYMMETRY Step 1: Identify a, b, c Step 2: Solve for

The AXIS OF SYMMETRY Find the axis of symmetry of a = 2, b = 4, c = 5

The AXIS OF SYMMETRY Find the axis of symmetry of a = –1, b = 1, c = 7

The VERTEX FORMULA This should not scare you: Step 1: Identify a, b, c Step 2: Solve for Step 3: Plug your answer back in to the original equation and solve for y = f(x). Step 4: Write your answer as an ordered pair.

SONG Axis of Symmetry and Vertex Formula Tune: “Mary Had a Little Lamb” To find the axis of symmetry Use x equals negative b Over 2a, don’t you see … It’s a vertical line! It splits the parabola in two. Helps you find the vertex too. Substitute for x is all you do. Solve for y and you’re just fine!

The AXIS OF SYMMETRY Find the vertex of a = 2, b = 4, c = 5

The AXIS OF SYMMETRY Find the vertex of a = – 1, b = –2, c = 0

Find the Vertex