CH 5 PT 1 PORTFOLIO PAGE – 5-1
THROUGH
5-5
Lesson 5-1: Modeling Data with Quadratic Functions
--identify quadratic functions & their graph --Quadratic regression on graphing calculator
STANDARD FORM OF QUADRATIC FUNCTION: y = ax2 + bx + c
Quadratic term: ax2 Linear term: bx Constant term: c
Example: Determine if the function is linear or quadratic. State the quadratic, linear, and constant terms:
1) f(x) = 3x2 – (x + 3)(2x – 1)
f(x) = x2 – 5x + 3; quadratic/ x2 / -5x / 3
GRAPH OF A QUADRATIC FUNCTION:
Parabola – the graph of a quadratic function
(U-shaped)
Axis of Symmetry -- the line that divides a
parabola into two parts that are mirror
images.
Vertex – the point at which the parabola
intersects the axis of symmetry.
If graph opens up, vertex is MINIMUM. If
graph opens down, vertex is MAXIMUM.
P
Q
Example:
1) State
2) State
3) State
4) State
(4, 1)
vertex:
X=4
axis of symmetry:
P’
(2, 0)
Q’
(8, -3)
QUADRATIC REGRESSION ON GRAPHING CALCULATOR: Write a quadratic model given points on the graph.
In calculator:
1) STAT / 1:Edit – enter x values in L1 and y values in L2
2) STAT / CALC / 5:QuadReg
3) VARS / Y-VARS / 1:Function / 1:Y1
4) ENTER
Substitute values of a, b, and c into standard form.
Example: Find the quadratic model for the values (-1, 10), (2, 4), (3, -6)
Y = -2x2 + 12
The quadratic function in standard form is ____________________________________
Lesson 5-2: Properties of Parabolas
--graphing from standard form --finding minimum and maximum values of quadratic functions
GRAPH FROM STANDARD FORM:
f(x) = ax2 + bx + c:
1)
2)
If a is (+), parabola opens up.
If a is (-), parabola opens down.
3)
Vertex: (
4)
5)
−𝒃
−𝒃
, 𝒇( ))
𝟐𝒂
𝟐𝒂
Axis of symmetry: x =
Y-intercept: (0, c)
Graph:
Ex: y =
(y = ax2 + c)
-x2
+ 4
V: (0, 4)
Pts: (1, 3);(2, 0)
Graph:
(y = ax2 + bx + c)
Ex: y = x2 + 2x - 6
V: (-1, -7))
Pts: (0, -6);(1, -3)
−𝒃
𝟐𝒂
ALL GRAPHS SHOULD INCLUDE:
--axis of symmetry
--vertex
--two pts to left and right of vertex
M. MURRAY
VERTEX FORM
OF
Lesson 5-3: Translating Parabolas
--Using Vertex Form to graph and write equations
QUADRATIC FUNCTION: y = a(x – h)2 + k
Vertex: (h, k)
Axis of Symmetry: x = h
Graph:
y = -3(x+1)2 + 4
V: (-1, 4)
Pts: (0, 1);(1, -8)
WRITING EQUATIONS: Given vertex and one point
Example: Vertex: (-2, 5) Point: (3, 4)
Y=-
1
(𝑥
25
+ 2)2 +5
IDENTIFY
VERTEX AND Y-INTERCEPT FOR EACH FUNCTION:
Example (standard form)
Example (vertex form)
Y = -3x2 + 6x – 1
y = (x – 2)2 + 3
V: (2, 3)
Y-int: (0, 7)
V: (1, 2)
Y-int: (0, -1)
STANDARD FORM/VERTEX FORM:
Convert the function to standard form:
y = 2(x –
3)2
Convert the function to vertex form:
+ 4
y = 2x2 – 4x + 3
Y = 2x2 + 6x + 4
Y = 2(x – 1)2 + 1
Lesson 5-5: Solving Quadratic Equations
--by factoring, graphing, and square roots
1)
2)
3)
4)
TO SOLVE QUADRATIC EQUATIONS BY FACTORING:
Write equations in standard form (set = to zero)
Factor
Apply zero product property and set each variable factor to zero.
Solve the equations
TO SOLVE BY FINDING SQUARE ROOTS:
1) Isolate squared term on one side of equation
2) Take the square root of each side. *don’t forget
1) x2 = 16x – 48
x = 12, x = 4
2) 9x2 – 16 = 0
4
x = ±3
±
TO SOLVE BY GRAPHING:
1) Graph the related function y = ax2 + bx + c
2) Find ZEROS (x-intercepts):
2nd/CALC/Zero
Left bound, Right bound, Guess
3)
x2 – 5x + 2 = 0
M. MURRAY