Download presentation

1
**5.1 Modeling Data with Quadratic Functions**

Quadratic Functions and Their Graphs

2
**1) Quadratic Formulas and Their Graphs**

A quadratic function is a function that produces a parabola.

3
**1) Quadratic Formulas and Their Graphs**

A quadratic function is a function that produces a parabola.

4
**1) Quadratic Formulas and Their Graphs**

A quadratic function is a function that produces a parabola.

5
**1) Quadratic Formulas and Their Graphs**

The equation of a quadratic function can be written in standard form. Quadratic term Linear term Constant term

6
**Quadratic Function: f(x) = ax2 + bx + c ‘a’ cannot = 0**

7
**1) Quadratic Formulas and Their Graphs**

Since the largest exponent of function is 2, we say that a quadratic equation has a degree of 2. Equations of second degree are called quadratic.

8
**QUADRATICS - - what are they?**

Y = ax² bx c FORM _______________________ Quadratic term Linear term Constant Important Details c is y-intercept a determines shape and position if a > 0, then opens up if a < 0, then opens down Vertex: x-coordinate is at –b/2a

9
**Parts of a parabola These are the roots Roots are also called: -zeros**

-solutions - x-intercepts This is the y-intercept, c It is where the parabola crosses the y-axis This is the vertex, V This is the called the axis of symmetry, a.o.s. Here a.o.s. is the line x = 2

10
**STEPS FOR GRAPHING Y = ax² + bx +c**

HAPPY or SAD ? 2 VERTEX = ( -b / 2a , f(-b / 2a) ) T- Chart Axis of Symmetry

11
**GRAPHING - - Standard Form (y = ax² + bx + c) y = x² + 6x + 8**

The graph will be symmetrical. Once you have half the graph, the other two points come from the mirror of the first set of points. 1) It is happy because a>0 FIND VERTEX (-b/2a) a =1 b=6 c=8 So x = -6 / 2(1) = -3 Then y = (-3)² + 6(-3) + 8 = -1 So V = (-3 , -1) 3) T-CHART X Y = x² + 6x + 8 Why -2 and 0? Pick x values where the graph will cross an axiw -2 y = (-2)² + 6(-2) + 8 = 0 y = (0)² + 6(0) + 8 = 8 11

12
**GRAPHING - - Standard Form (y = ax² + bx + c) y = -x² + 4x - 5**

1) It is sad because a<0 FIND VERTEX (-b/2a) a = b=4 c=-5 So x = - 4 / 2(-1) = 2 Then y = -(2)² + 4(2) – 5 = -1 So V = (2 , -1) Here we only have one point where the graph will cross an axis. Choose one other point (preferably between the vertex and the intersection point) to graph. 3) T-CHART X Y = -x² + 4x - 5 1 y = -(1)² + 4(1) - 5 = -2 y = -(0)² + 4(0) – 5 = -5 12

13
**1) Quadratic Formulas and Their Graphs**

Example 1: Determine whether each function is linear or quadratic. Identify the quadratic term, linear term and constant term.

14
**1) Quadratic Formulas and Their Graphs**

Example 1: Determine whether each function is linear or quadratic. Identify the quadratic term, linear term and constant term. This IS a quadratic function. QUADRATIC TERM: x2 LINEAR TERM: 3x CONSTANT TERM: none

15
**1) Quadratic Formulas and Their Graphs**

Example 1: Determine whether each function is linear or quadratic. Identify the quadratic term, linear term and constant term. This IS a quadratic function. QUADRATIC TERM: x2 LINEAR TERM: 3x CONSTANT TERM: none This is NOT a quadratic function. QUADRATIC TERM: none LINEAR TERM: 5x CONSTANT TERM: none

16
EX 3 Find the vertex, axis of symmetry and the corresponding points to P and Q. y = x2 – 6x + 11

17
Ex 1 Is the function linear or quadratic? f(x) = (2x – 1)2

18
EX 2 Is the function linear or quadratic? f(x) = x2 – (x + 1)(x – 1)

19
**EX 4 Find a quadratic function to model the given points:**

(-2, -17) (1, 10) (5, -10)

20
Ex 5 y = 2x2 + x – c contains the point (1, 2). Find c.

21
**1) Quadratic Formulas and Their Graphs**

We can graph parabolas using a table of values.

22
**1) Quadratic Formulas and Their Graphs**

We can graph parabolas using a table of values. Recall…graphing linear functions…

23
**1) Quadratic Formulas and Their Graphs**

Example 2: Graph the parent function f(x) = x2 using a table of values.

24
**1) Quadratic Formulas and Their Graphs**

Example 2: Graph the parent function f(x) = x2 using a table of values. x y -2 -1 1 2

25
**1) Quadratic Formulas and Their Graphs**

Example 2: Graph the parent function f(x) = x2 using a table of values. x y -2 (-2)2 = 4 -1 (-1)2 = 1 (0)2 = 0 1 (1)2 = 1 2 (2)2 = 4

26
**1) Quadratic Formulas and Their Graphs**

Example 2: Graph the parent function f(x) = x2 using a table of values. x y -2 4 -1 1 2

27
**1) Quadratic Formulas and Their Graphs**

The axis of symmetry is a line that divides the parabola in half.

28
**1) Quadratic Formulas and Their Graphs**

The axis of symmetry is a line that divides the parabola in half. The vertex is a maximum or minimum of the parabola.

29
**1) Quadratic Formulas and Their Graphs**

The axis of symmetry here is x = 0 The vertex here is a minimum at (0, 0)

30
**1) Quadratic Formulas and Their Graphs**

Points on the parabola have corresponding points that are equidistant from the axis of symmetry. A B A’ B’

31
**1) Quadratic Formulas and Their Graphs**

Example 3: Identify the vertex and axis of symmetry for the parabola. Identify points corresponding to P and Q. 3 P 2 1 -2 -1 1 Q 2 3 4 -1 -2

32
**1) Quadratic Formulas and Their Graphs**

Example 3: Identify the vertex and axis of symmetry for each parabola. Identify points corresponding to P and Q. 3 Vertex: (1, -1) Axis of symmetry: x = 1 P’ (3, 3) Q’ (0, 0) P P P’ 2 1 -2 -1 1 Q Q’ 2 3 4 -1 -2

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google