5.1 modeling data with quadratic functions 1.quadratic functions and their graphs
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5.1 Modeling Data with Quadratic Functions
1. Quadratic Functions and Their Graphs
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.
-2
-1
0
1
2
3
4
-3 -2 -1 0 1 2 3
1) Quadratic Formulas and Their Graphs
The equation of a quadratic function can be written in standard form.
cbxaxxf 2)(
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?
Important Detailso c is y-intercepto a determines shape and position
if a > 0, then opens up
if a < 0, then opens downo Vertex: x-coordinate is at –b/2a
FORM _______________________Y = ax² + bx + c Quadratic
term
Linear
term
Constant
Parts of a parabola
This is the y-intercept, cIt is where the parabola crosses the y-axis
This is the vertex, V
This is the calledthe axis of symmetry, a.o.s.Here a.o.s. is the line x = 2
These are the rootsRoots are also called:-zeros-solutions- x-intercepts
STEPS FOR GRAPHINGY = ax² + bx +c
1 HAPPY or SAD ?2 VERTEX = ( -b / 2a , f(-b / 2a) )3 T- Chart4 Axis of Symmetry
GRAPHING - - Standard Form (y = ax² + bx + c)
y = x² + 6x + 8 1) It is happy because a>0
2) 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
-2 y = (-2)² + 6(-2) + 8 = 0
0 y = (0)² + 6(0) + 8 = 8
Why -2 and 0?
Pick x values where the graph will cross an axiw
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.
GRAPHING - - Standard Form (y = ax² + bx + c)
y = -x² + 4x - 5 1) It is sad because a<0
2) FIND VERTEX (-b/2a) a =-1 b=4 c=-5 So x = - 4 / 2(-1) = 2 Then y = -(2)² + 4(2) – 5 = -1
So V = (2 , -1)
3) T-CHART
X Y = -x² + 4x - 5
1 y = -(1)² + 4(1) - 5 = -2
0 y = -(0)² + 4(0) – 5 = -5
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.
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.
)3()() xxxfa 22 )5()() xxxxfb
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.
xx
xxxfa
3
)3()()2
22 )5()() xxxxfb
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.
xx
xxxfa
3
)3()()2
x
xxx
xxxxfb
5
5
)5()()22
22
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
0
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 (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
0 0
1 1
2 4
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.
-2
-1
0
1
2
3
4
-2 -1 0 1 2 3 4
P
Q 4321-1
-2
-2 -1
1
2
3
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.
P Vertex: (1, -1)
Axis of symmetry: x = 1
P’ (3, 3)
Q’ (0, 0)
-2
-1
0
1
2
3
4
-2 -1 0 1 2 3 4
P
QQ’
P’
4321-1
-2
-2 -1
1
2
3