Download Quadratic Functions

Algebra 2
Quadratic Functions Section 6-1/6-2 Day 1
Learning Target:
1. Graph quadratic functions
2. Find and interpret the maximum and minimum values of a quadratic function.
Quadratic Function: f ( x)  ax 2  bx  c
Quadratic term
where a  0
Linear term
Constant term:
Parabola: The graph of a quadratic function.
Minimum Point: (a is positive)
a  0 opens upward
Maximum point: (a is negative)
a  0 opens downward
b
if fold the parabola along its axis of symmetry, the
2a
portions of the parabola on either side of this line would match.
Axis of Symmetry: x 
  b   b 
, f
Vertex: 
  the point where the axis of symmetry intersects the parabola.
 2a  2a  
Y-intercept, let x=0. OR (0,c)
X-Intercepts – Root sof the equations – Solutions- Zeros of the function (let y = 0):
Sketch each:
No solution:
One solution:
Two solutions:
Example #1:
What kind of an equation is y = x + 3?
What kind of an equation is y = x2 + 3?
Linear or quadratic
Linear or quadratic
What is the degree of x? _________
What is the degree of x? _________
Any polynomial that is written in the form 0= ax2+bx+c is what we call a Quadratic
Equation.
Any polynomial that is written in the form f(x)= ax2+bx+c where a  0 is what we call a
Quadratic Function.
These are quadratics.
3x2 + 4x + 1 = 0
3x2 + 1 = y
4y + ½ x2 = 14
7 + x2 = 4y + 12
These are not quadratics.
x2 + y2 – 2xy = 12
y = 2x + 2
y
1
x2
x3 = y
Example #2
Please identify which of the following are linear(L), quadratics(Q), or neither(N).
2x + 4y = 7
x2 + y2 = 17
2x2 + 4x + 12 = y
2x2 + y + 12 = 0
2x5 + 4x2 = y
Example #3
Show V-vertex(plot a point), AOS-axis of symmetry(draw the vertical line), state if it is a
max or min, place a point(s) for the x-intercepts, place a point for the y-intercept.(label)
5 MAJORS STEPS!!
Example #4:
1. f ( x)  x 2  2 x  8
a=
b=
c=
Step 1: Find the Axis of Symmetry(AOS): Formula: x 
b
2a
Process:
Use the formula:
Write answer x=
Step 2: Find the vertex:
Process:
Find AOS x= (from step 1)
Plug in the (AOS x= ) into the function to find y  f (x) , the vertex is a point (x,y)
Write answer as a point(x,y)
The y coordinate is also the maximum or minimum value of the function.
Step 3: Determine if it is a maximum or minimum
Look at a.
If a is positive, then it is a minimum
If a is negative, then it is a maximum
Step 4: Find the x and y intercepts
y intercept is (0,c) if in the form f ( x)  ax  bx  c
2
if not, let x=0
x-intercept-Roots-Zeros-solutions: factor or use the calculator
Step 5: Make a table and Graph
Include the vertex, y intercept, x-intercept, and points on either side of the vertex.
Use symmetry to help find other points on the graph of the parabola, think of a mirror
image from the axis of symmetry. Use the calculator and the table to help as well!
x
y
Example #5: Calculator:
1. f ( x)  x 2  2 x  3
a=
b=
c=
Step 1: AOS
Process:
plug in y1  x 2  2 x  3 enter, zoom 6, determine if it is a maximum or minimum.
2nd trace, enter min or max enter,
Left bound, have the calculator in front of you, left side of the vertex is the left bound,
this should be to your left side of the graph, enter
Move to the right side of the graph, past the vertex, this is right bound. This should be to
the right side of the graph. Enter
Guess-hit enter again
Vertex is given:
AOS is the x-coordinate of the vertex, state this x=
Step 2: Find the vertex: already have it from above:
Step 3: Determine if it is a maximum or minimum-already have done this:
Step 4: Find the x and y intercepts
Process: 2nd window:
Tblstart: this is to start your table: =0
Tbl  1 change in the table
Auto
Auto
2nd graph:
X=0 y=y intercept
y=
x-intercept:
look at table where y=0. This will be the x-intercept. Use the up and down arrow to
search.
Step 5: Make a table and Graph: Which points to includex
y