This document discusses different forms of quadratic functions: vertex form, standard form, and intercept form. It provides examples of rewriting quadratic functions between these forms using techniques like completing the square, factoring, and FOIL. Key characteristics like the vertex, axis of symmetry, intercepts, and concavity are identified for each form. Students are asked to practice rewriting quadratic functions between forms and finding the important characteristics.

1. Forms of Quadratics
QUADRATIC FUNCTIONS-ALGEBRA 2
CHRISTINE GU

2. Lecture Objectives
After this lecture, students will be able to:
1. Recognize and graph a quadratic in standard form, vertex form, and intercept form
2. Identify important characteristics of a quadratic from its equation.
3. Transform a quadratic between its three forms by completing the square, multiplying,
or factoring
California Common Core Standards
F-IF.8 Write a function defined by an expression in different but
equivalent forms to reveal and explain different properties
of the function.

3. What Do You Remember?
Recall the parent function of a quadratic
function.
Discuss with your partner:
• What is the vertex of this functions? Axis of
symmetry?
• Which direction does this parabola open?
𝑓 𝑥 = 𝑥2

4. Vertex Form
Compared to the parent function 𝑔 𝑥 = 𝑥2, 𝑓 𝑥 has a:
• Horizontal shift of h units and a vertical shift of k units
• Vertical stretch/compression by a factor of a
• Reflection across the x-axis if a < 0
𝑓 𝑥 = 𝑎 𝑥 − ℎ 2
+ 𝑘

5. Vertex Form (cont.)
Vertex: ℎ, 𝑘
Axis of symmetry: 𝑥 = ℎ
y-intercept: 0, 𝑓 0
If a > 0, f(x) is concave up.
Otherwise if a < 0 , f(x) is concave down.
𝑓 𝑥 = 𝑎 𝑥 − ℎ 2 + 𝑘

6. Example
Let 𝑓 𝑥 = −2 𝑥 − 1 2
+ 3.
ℎ = 1,
𝑘 = 3,
𝑎 = −2
vertex: (1, 3)
AOS: x = 1
y-intercept: (0, 1)
f(x) is concave down
𝑓 0 = −2 0 − 1 2 + 3
= −2 −1 2 + 3
= −2 ∙ 1 + 3
= 1
2. Find the y-intercept:
1. Identify h, k, and a:
Graph of f(x) generated with desmos

7. Now You Try
Let 𝑓 𝑥 = 3 𝑥 − 5 2
− 4.
Work with your partner to identify the vertex, axis of symmetry, and y-
intercept of this quadratic equation. Then graph f(x) and label the
characteristics.

8. Vertex Form to Standard Form
Expand 𝑓 𝑥 = 𝑎 𝑥 − ℎ 2 + 𝑘 by FOILing and combining like terms.
from Friendly Math 101 on Youtube

9. Standard Form
Vertex:
−𝑏
2𝑎
, 𝑓
−𝑏
2𝑎
Axis of symmetry: x =
−𝑏
2𝑎
y-intercept: (0, c)
If a > 0, f(x) is concave up.
Otherwise if a < 0 , f(x) is concave down.
𝑓 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐

10. Example
Let f x = 3𝑥2 + 6𝑥 + 4.
𝑎 = 3, 𝑏 = 6, 𝑐 = 4
1. Identify a, b, and c:
Graph of f(x) generated with desmos
2. Find the vertex:
−𝑏
2𝑎
=
−6
2 ∙ 3
= −1
𝑓 −1 = 3 −1 2
+ 6 −1 + 4
= 3 1 − 6 + 4 = 1
vertex: (-1, 1)
AOS: x = -1
y-intercept: (0, 4)
f(x) is concave up

11. Standard Form to Vertex Form
Complete the square for 𝑓 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐.
from The Organic Chemistry Tutor on Youtube

12. Standard Form and Intercept Form
→ Factor 𝑓 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐.
Example
Let 𝑓 𝑥 = −2𝑥2 − 4𝑥 + 6. Work with your partner to f(x)
completely.
← multiply/FOIL 𝑓 𝑥 = 𝑎 𝑥 − 𝑝 𝑥 − 𝑞 .
𝑓 𝑥 = −2𝑥2 − 4𝑥 + 6
= − 2 x2 + 2x − 3
= −2(𝑥 + 3)(𝑥 − 1)

13. Now You Try
Let f x = x − 5 x − 1 .
Work with your partner to rewrite f(x) into standard form. Then identify the
vertex, axis of symmetry, and y-intercept of this quadratic equation.

14. Intercept Form
x-intercepts: (p, 0) and (q, 0)
vertex:
𝑝+𝑞
2
, 𝑓
𝑝+𝑞
2
Axis of symmetry: x =
𝑝+𝑞
2
y-intercept: 0, 𝑓 0
If a > 0, f(x) is concave up.
Otherwise if a < 0 , f(x) is concave down.
𝑓 𝑥 = 𝑎 𝑥 − 𝑝 𝑥 − 𝑞

15. Example
1. Identify a, p, and q:
Let 𝑓 𝑥 = −2(𝑥 + 3)(𝑥 − 1)
a = −2, p = −3, q = 1
2. Find the vertex:
𝑝 + 𝑞
2
=
−3 + 1
2
= −1
3. Find the y-intercept:
𝑓 0 = −2 0 + 3 0 − 1
= −2 3 −1 = 6
x-ints: (-3, 0) and (1, 0)
vertex: (-1, 8)
AOS: x = -1
y-intercept: (0, 6)
f(x) is concave down
f −1 = −2 −1 + 3 −1 − 1
= −2 2 −2 = 8
Graph of f(x) generated with desmos

16. Summary
Vertex Form
𝑓 𝑥 = 𝑎 𝑥 − ℎ 2 + 𝑘
• v: (h, k)
Standard Form
𝑓 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐
• y-intercept: (0, c)
• v:
−𝑏
2𝑎
, 𝑓
−𝑏
2𝑎
Intercept Form
𝑓 𝑥 = 𝑎 𝑥 − 𝑝 𝑥 − 𝑞
• x-intercepts: (p, 0), (q, 0)
• v:
𝑝+𝑞
2
, 𝑓
𝑝+𝑞
2
Expand Factor
Complete the square Multiply/FOIL

17. Now You Try
Let 𝑓 𝑥 = −𝑥2 + 4𝑥 + 5.
a. What form is this? Rewrite f(x) into the other two forms.
b. Identify the vertex, axis of symmetry, x- and y-intercepts, and
concavity.
c. Graph and label f(x).