Lesson 3.2: Quadratic Functions and Graphs
COMPLETING THE SQUARE-GRAPHING QUADRATICS-VERTEX FORMULA-EXTREME VALUES-APPLICATIONS
Warm-Up
Consider the parent function y = x 2 .
1) Describe the transformation, given the image y = 2(x – 4) 2 + 3
Horizontal Shift: Vertical Stretch/Shrink: Vertical Shift:
2) Sketch the parent function AND image:
Step 1: Parent Step 2: Horizontal Shift
Step 3: Vertical Stretch Step 4: Vertical Shift
Lesson Starts HERE!
Quadratic Functions: f(x) = ax 2 + bx + c
If a > 0 :
________________________________________________________________________
If a <
0:_________________________________________________________________________
(0, c) is the ___________ intercept
How do I find the vertex of an equation in standard form?
The x-coordinate of the vertex can be found by: x = _______________
To find the y – coordinate, substitute the x-value into your equation.
Example: Find the vertex of y = 2x 2 + 8x – 3. Is this a minimum or maximum? How do you know?

Lesson 3.2: Quadratic Functions and Graphs
COMPLETING THE SQUARE-GRAPHING QUADRATICS-VERTEX FORMULA-EXTREME VALUES-APPLICATIONS
How do I use the vertex formula to graph the function?
Graph y = 3x 2 + 6x + 1
Step 1: Find the vertex
VERTEX: (______________, ______________)
Step 2: Make a table, putting the vertex in the MIDDLE of the table and filling in the integer xvalues surrounding it. Use your CALCULATOR to fill out the table. x y
Step 3: Graph the points.
Vertex Form: f(x) = 𝒂(𝒙 − 𝒉) 𝟐 + 𝒌
(h, k) represents the________________________________.
Perfect Square Trinomials:
(a + b) 2 = a 2 + 2ab + b 2
WHY?
Example: Factor x 2 + 6x + 9
(a – b) 2 = a 2 – 2ab + b 2

Lesson 3.2: Quadratic Functions and Graphs
COMPLETING THE SQUARE-GRAPHING QUADRATICS-VERTEX FORMULA-EXTREME VALUES-APPLICATIONS
To complete the square y = x 2 – 4x – 6
Parent: Image:
Move the constant towards the y.
Complete the square by adding ( ½ b) 2 to both sides
Factor the side of the equation with x’s
Isolate the y
What is the vertex?
Sketch the graph, based on the translation.
What if there is a coefficient of x 2 ? y = 2x 2 + 4x - 16
Move the constant towards the y.
Divide both sides by the coefficient of x 2
Complete the square by adding ( ½ b) 2 to both sides
Factor the side of the equation with x’s
Isolate the y
What is the vertex?
What is the vertical stretch/shift?

End of Day 1!
Lesson 3.2: Quadratic Functions and Graphs
COMPLETING THE SQUARE-GRAPHING QUADRATICS-VERTEX FORMULA-EXTREME VALUES-APPLICATIONS
(1) Parent (2) Horizontal Shift
(3) Vertical Stretch (4) Vertical Shift
Closure: For the given quadratic function y = 3x 2 +3x - 6
Sketch the graph, based on the transformation.
1) Write in vertex form: y = a(x – h) 2 + k
2) Give the vertex of the parabola
3) Graph the function

Lesson 3.2: Quadratic Functions and Graphs
COMPLETING THE SQUARE-GRAPHING QUADRATICS-VERTEX FORMULA-EXTREME VALUES-APPLICATIONS
Warm-Up
For the given quadratic function y = 3x 2 +4x - 1
1) Write in vertex form: y = a(x – h) 2 + k
2) Give the vertex of the parabola
3) Graph the function
Lesson: Applications and Quadratic Models
Student Exploration
Enter the data in the calculator and determine a quadratic equation.
Year (time after
1970)
Enrollment (in thousands)
0
447
10
376
20
541
36
909
1) STAT, EDIT, L1 = time and L2 = enrollment
2) Turn your plots ON(2 nd , y = ). ZOOM 9 lets you see the data. Sketch what you see below:
3) STAT, CALC, 5:QuadReg gives the quadratic model. Write it down here:
4) Now, find the vertex of your model, using what we learned in the previous
class (x = −𝑏
2𝑎
). What does the vertex mean?

Lesson 3.2: Quadratic Functions and Graphs
COMPLETING THE SQUARE-GRAPHING QUADRATICS-VERTEX FORMULA-EXTREME VALUES-APPLICATIONS
Height of a Projectile Object (Feet Only) s(t) = -16t 2 + v
0 t + s
0 s(t) = ___________________ t = ___________________ v
0
= ___________________ s
What are some examples of “projectiles?”
0
= ___________________
Example: A projectile is shot from a tower 10 feet high with an upward velocity of 100 feet/second.
1) Approximate the relationship between height (in feet) and time(in seconds) after the projectile is shot.
2) What is the vertex, and what does it mean in the context of this situation?
3) How long will the projectile be in the air? (What will the height be at the end of the flight?)
Using the calculator: Let y
1
= equation and y
2
= ________
Adjust your window by looking at the table. (Remember, turn any plots OFF now)
2 nd , TRACE, INTERSECT, ENTER, ENTER, ENTER
4)
When will the projectile reach 37 feet for the first time
?
Using the calculator: Let y
1
= equation and y
2
= ________
**You should move your cursor closer to that intersection!
2 nd , TRACE, INTERSECT, ENTER, ENTER, ENTER
5) What is the domain and range of this function?