 # 5.1 Modeling Data with Quadratic Functions

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5.1 Modeling Data with Quadratic Functions

1) Quadratic Formulas and Their Graphs
A quadratic function is a function that produces a parabola.

1) Quadratic Formulas and Their Graphs
A quadratic function is a function that produces a parabola.

1) Quadratic Formulas and Their Graphs
A quadratic function is a function that produces a parabola.

1) Quadratic Formulas and Their Graphs
The equation of a quadratic function can be written in standard form.  Quadratic term Linear term Constant term

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

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.

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

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

STEPS FOR GRAPHING Y = ax² + bx +c
HAPPY or SAD ? 2 VERTEX = ( -b / 2a , f(-b / 2a) ) T- Chart Axis of Symmetry

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

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

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.

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

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

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

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

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

EX 4 Find a quadratic function to model the given points:
(-2, -17) (1, 10) (5, -10)

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

1) Quadratic Formulas and Their Graphs
We can graph parabolas using a table of values.

1) Quadratic Formulas and Their Graphs
We can graph parabolas using a table of values. Recall…graphing linear functions…

1) Quadratic Formulas and Their Graphs
Example 2: Graph the parent function f(x) = x2 using a table of values.

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

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

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

1) Quadratic Formulas and Their Graphs
The axis of symmetry is a line that divides the parabola in half.

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.

1) Quadratic Formulas and Their Graphs
The axis of symmetry here is x = 0 The vertex here is a minimum at (0, 0)

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’

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

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