1.7 Graphing Quadratic Functions

1. Find the x-intercept(s).

• The x-intercepts occur when • Solve by:• Factoring• Completing the Square• Quadratic Formula

y = 0

We will get to this!

Use substitution!!

2. Find the y-intercept.

• The y-intercept occurs when •When the function is in the form y = ax2 + bx + c,

x = 0

c is the y-intercept.

Use substitution!!

3. Find the axis of symmetry.

• The line is the axis of symmetry.• It is a vertical line.

4. Find the vertex.• is the x-coordinate of the vertex.• Plug into your equation (y = ax2 + bx +

c) to find the y-coordinate of the vertex.• (You are really just solving a very basic

system of equations to find the intersection between a vertical line and a parabola.)

4. Finding the vertex

y = ax2 + bx + c

𝑥=− 𝑏2𝑎

4. Finding the vertex.• If your function is written in function

notation...f(x) = ax2 + bx + c

…then the vertex can be expressed as:

( , )

4. Finding the vertex

y = ax2 + bx + c

𝑥=− 𝑏2𝑎

(− 𝑏2𝑎 , 𝑓 (− 𝑏

2𝑎 ))

Practice: Graph the following quadratic functions.

1. y = 2x2 – 7x + 5

2. y = –2x2 + 12x + 4

Factored Form•What if we had a quadratic that is

factored?•Consider y = (2x-1)(x-3)•What’s the vertex?

Factored Form•Consider y = (2x-1)(x-3)•What’s the vertex?• Find the x-intercepts.

Factored Form•Consider y = (2x-1)(x-3)•What’s the vertex?• Find the x-intercepts.•Now average them! This is your x-

value for your vertex.• Find the y-value.

Predicting what the graph will look like…

•If a > 0, the graph opens upward

•If a < 0, the graph opens downward

Vertex Form of a Quadratic Functiony = a(x – h)2 + k

• The vertex is (h, k).• a is just some constant• Ex: y = 2(x – 3) 2 + 7• The vertex is (3, 7)

• Ex: y = –4(x + 9) 2 + 2 => y = –4(x – –9) 2 + 2• The vertex is (–9, 2)

Graphing in Vertex Form

Example: y = 3(x – 2)2 – 9

1. Find the vertex.y = 3(x – 2)2 – 9

•y = 3(x – 2)2 + –9•The vertex is (2, –9)

2. Find the axis of symmetry.

• The axis of symmetry will always be the vertical line:

x = (the x-coordinate of the vertex)

• For the function y = 3(x – 2)2 – 9, the axis of symmetry is:

x = 2

3. Find the y-intercept.

y = 3(x – 2)2 – 9

• The y-intercept happens when x = 0.• y – intercept = 3

4. Find the x-intercept.

y = 3(x – 2)2 – 9

• The x-intercept happens when y = 0.• x – intercept = or x ≈ 3.7, 0.3

Practice:

•Graph y = (x+2)2 – 3

Practice:

•Graph y = (x+2)2 – 3

Practice:

•Graph y = –2(x – 2)2

Practice:

•Graph y = –2(x – 2)2