Download Pre-AP Algebra 2 Unit 3 – Lesson 1 – Quadratic

Pre-AP Algebra 2
Unit 3 – Lesson 1 – Quadratic Functions
Objectives: The students will be able to
 Identify and sketch the quadratic parent function
 Identify characteristics including vertex, axis of symmetry, x-intercept, and y-intercept
 Translate the quadratic function using a (stretch), h (horizontal) and k (vertical)
 Identify domain and range of quadratic functions
Materials: Little Black Book; Do Now worksheet; pairwork; hw #3-1
Time
Activity
Pass out new Little Black Book
15 min
Do Now
Hand out the Graphing Parabolas sheet to students. First they compare the parent function for
( )
, to understand that is not the same as slope, but does affect the steepness of the graph.
They are given a parabola in vertex form, and must identify the vertex and x-intercepts, which way the
parabola is opening and its width relative to the graph of y = x2, and then make a graph. They also have
one intercept form parabola. They must make an x-y table to graph it. Review the problems, and
discuss how they could use the given intercept form function to easily find the x-intercepts and the
vertex (i.e. the midpoint of the x-intercepts).
30 min
Direct Instruction
Background Information:
1) Identify the parts of a parabola on the given example: vertex, axis of symmetry, x-intercepts (roots)
2) Determine the number of possible roots of a parabola: where does f(x), g(x), and h(x) = 0?
3) To find the roots, set f(x) = 0 and solve.
Concepts:
There are three forms a quadratic equation can be written in:
1) Vertex Form 2) Intercept Form
3) Standard Form
Vertex Form: f(x) = a(x – h)2 + k
 Comes from translating and transforming the function f(x) = x2 to f(x) = a(x – h)2 + k.
 x-intercepts: set equal to 0 and solve by working backward
 vertex: (h, k)
Intercept Form: f(x) = a(x – m)(x – n)
 This can also be called “factored form”.
 x-intercepts: set each factor equal to 0 and solve.
 vertex: halfway between the x-intercepts; find the average:


mn  mn 


 2 , f  2  
Standard Form: f(x) = ax2 + bx + c
 x-intercepts: set equal to 0, and solve by factoring or using quadratic formula

vertex: find average if you factored it; otherwise, use the formula


 b   b
 , f  
 2a  2a  
.
For all Forms:
 If a is positive, the parabola opens up; if a is negative, the parabola opens down
 If |a| > 1, the parabola is steeper than y = x2; if 0< |a| < 1, the parabola is flatter than y = x2
To Graph Any Parabola:
 Plot the vertex (and x-intercepts, if they are integers).
 Draw a dotted axis of symmetry (lightly).
 Plot the y-intercept f(0) and its reflection.
 If you need more points, pick x-values on one side of the axis and plug them in to the function.
Plot them, and their reflections.
Pre-AP Algebra 2
Lesson #3-1: Do Now
Name: _______________________
Graphing Parabolas
1. Fill in the table for ( ) and ( ). Then, graph each function on the axes.
( )
( )
-3
-2
-1
0
1
2
3
Compare ( ) and ( )
( )
( )
Vertex
Axis of Symmetry
Domain
Range


2
2. Given the function f (x)  2 x  3  8 ,
a. Describe the translation and transformation from y  x 2 .
b. What is the vertex of the graph of f (x) ?
c. Does the graph of f (x) open up or down?
d. Is it steeper or flatter than the graph of y  x 2 ?
e. Find the x-intercepts of the graph by solving f (x)  0 .
Pre-AP Algebra 2
Lesson #3-1: Do Now
Name: _______________________
f. Use the previous work to make a graph of f (x) .
g. Compare ( ) to the parent function.
( )
(
)
Vertex
Axis of Symmetry
Domain
Range
intercepts
intercept
(
)
h. Generalize: Given a function in the form ( )
Vertex:
Axis of Symmetry:
Domain:
How do you find the
Range:
intercepts and
intercept?
Pre-AP Algebra 2
Lesson #3-1: Do Now
Name: _______________________
3. Given the function ( )
(
)(
),
a. Find the x-intercepts by solving f (x)  0 .
b. Make an x-y table so that you can graph the parabola. Use the x-intercepts to figure out
what x-values to put in the table. Make sure the table goes beyond the x-intercepts.
c. Draw the graph of f (x) .
d. Which way does the graph open? Is it steeper or flatter than the graph of
?
e. Use the graph to determine the vertex of f (x) .
f. Explain how you can find the vertex without first graphing the parabola. See problem
4 for a hint if you are stuck.
4. Plot the two numbers on a number line. Then, determine the number exactly halfway
between the two numbers.
a. 3 and 9
b. 2 and 15
Pre-AP Algebra 2
Lesson #3-1: Homework
Name: _______________________
HW #3-1: Graphing Parabolas Practice
Check for Understanding
Can you complete these problems correctly by yourself
For each function, determine the following information:
1. The form the function is written in.
2. Its shape compared to the graph of y = x2 (opens up/opens down; steeper/flatter)
3. The vertex
4. The x-intercepts
Then, graph the parabola. Make sure to review your notes before you begin.


2
1) f (x)  3 x  2  75
1.
2.
3.
4.
2) f (x)  2(x  5)(x  4)
1.
2.
3.
4.
Pre-AP Algebra 2
Lesson #3-1: Homework
Name: _______________________
3) f (x)  x 2  3x  10
1.
2.
3.
4.
4) f (x)  2x 2  8
1.
2.
3.
4.
Spiral
What do you remember from Algebra 1and our previous units? (these are skills we will need
in this unit)Work on a separate sheet a paper
1. Given ( )
and ( )
. Find the following
)( )
a. (
d. ( ( ))
b. [ ( )]
e. ( ( ))
c. [ ( )]
2. Factor the following trinomials
a.
b.
c.
d.
Pre-AP Algebra 2
Lesson 3-1 –Notes
Concepts
There are three forms a quadratic equation can be
written in:
Name:________________________
Examples
Graph each function:

Background Information
Parts of a parabola:

2
1) f (x)  2 x  3  2
Vertex Form:

Info:

Vertex:

x-intercepts:
Number and types of x-intercepts:
Intercept Form:

Info:

Vertex:

x-intercepts:
Standard Form:

Vertex:
How to find the x-intercepts (roots):

x-intercepts:
Pre-AP Algebra 2
Lesson 3-1 –Notes
Name:________________________
Concepts
2) f (x)  (x  3)(x  6)
Shape of the Parabola (for all forms):
To Graph a Parabola:
1. Plot the vertex (and x-intercepts, if they are
integers).
2. Draw a dotted axis of symmetry (lightly).


3. Plot the point 0, f 0 (the y-intercept) and its
reflection.
4. If you need more points, pick x-values on one
side of the axis and plug them in to the
function. Plot them, and their reflections.
3) f (x) 
1 2
x  3x  4
2
Examples