C HAPTER properties of quadratic functions

O BJECTIVES Students will be able to: Define, Identify, and graph quadratic equations. Identify and use maximum and minimum of quadratic functions to solve problems

L AST TIME When you transformed quadratic functions in the previous lesson, you saw that reflecting the parent function across the y-axis results in the same function. This shows that parabolas are symmetric curves.

A XIS OF SYMMETRY What is an axis of symmetry? Answer: Is the line that goes through the vertex of a parabola and divides the parabola into two equal parts.

H OW DO WE IDENTIFY THE AXIS OF SYMMETRY

E XAMPLE 1

E XAMPLE 2

S TUDENT PRACTICE Identify the parabola's axis of symmetry ofparabola's axis of symmetry y =x² −2x −3 Solution:

E XAMPLE 3

S TUDENT P RACTICE What is the following parabola's axis of symmetry of y = (x + 3)² + 4.parabola's axis of symmetry Solution:

S TUDENT GUIDE Work of worksheet problems 1-4

S TANDARD FORM Another useful form of writing quadratic functions is the standard form. The standard form of a quadratic function is f(x)= ax 2 + bx + c, where a ≠ 0. The coefficients a, b, and c can show properties of the graph of the function. You can determine these properties by expanding the vertex form.

S TANDARD FORM f(x)= a(x – h) 2 + k f(x)= a(x 2 – 2xh +h 2 ) + k f(x)= a(x 2 ) – a(2hx) + a(h 2 ) + k f(x)= ax 2 + (–2ah)x + (ah 2 + k)

S TANDARD FORM a=a a in standard form is the same as in vertex form. It indicates whether a reflection and/or vertical stretch or compression has been applied.

S TANDARD FORM b =–2ah Solving for h gives. Therefore, the axis of symmetry, x = h, for a quadratic function in standard form is. c = ah 2 + k Notice that the value of c is the same value given by the vertex form of f when x = 0: f(0) = a(0 – h) 2 + k = ah 2 + k. So c is the y- intercept.

P ROPERTIES OF QUADRATIC FUNCTIONS These properties can be generalized to help you graph quadratic functions.

G RAPHING QUADRATIC EQUATIONS Consider the function f(x) = 2x 2 – 4x + 5. Graph that function using the properties. Solution: a. Determine whether the graph opens upward or downward. Because a is positive, the parabola opens upward. b. Find the axis of symmetry. The axis of symmetry is the line x = 1. c. c. Find the vertex. The vertex lies on the axis of symmetry, so the x-coordinate is 1. The y-coordinate is the value of the function at this x-value, or f(1).

G RAPHING QUADRATIC EQUATIONS d. Find the y-intercept. Because c = 5, the intercept is 5. Then graph

S TUDENT GUIDED PRACTICE Do problems 5 and 6 from worksheet. Use the properties.

F INDING THE MAXIMUM AND MINIMUM What is the minimum value? When the parabola opens upward, the y-value of the vertex is the minimum value. What is the maximum value? When the parabola opens downward the y-value of the vertex is the maximum value.

M AXIMUM AND MINIMUM VALUES

Find the minimum or maximum value of f(x) = –3x 2 + 2x – 4. Then state the domain and range of the function. Solution: Step 1 Determine whether the function has minimum or maximum value. Because a is negative, the graph opens downward and has a maximum value. Step 2 Find the x-value of the vertex.

CONTINUE Step 3 Then find the y-value of the vertex,

CONTINUE

S TUDENT GUIDED PRACTICE Do problems 11-16

H OMEWORK Do odd problems page 72 from book.

CLOSURE Today we learned about the properties of quadratic functions and we can use them to graph the function. Next class we are going to continue seeing quadratic equations but we are going to factor them.