1. General Form to Standard Form OR Vertex Form
By Completing the Square

2. General Form of Conics

3. 𝐴𝑥 2 + 𝐵𝑦 2 + Cx + Dy + E = 0
This term complicates things… we’ll leave it for Precal.
All conic sections can be written in the general form Ax2 + Bxy + Cy2 + Dx + Ey+ F = 0.
All conic sections can be written in the general form Ax2 + Bxy + Cy2 + Dx + Ey+ F = 0.
How many
squared terms?
Are
and
both present?
Do they
have the
same sign?
Do they have the same coefficients?
two
yes
yes
Circle
one no
Parabola no
Hyperbola
Ellipse

4. Circle Problem: Step 1 ---Rearrange the equation:
---x terms together on the left with a blank
---y terms together on the left with a blank
---constants on the right with 2 blanks
Step 2
---Fill in the blanks:
---half of the x term squared goes in the x blank
and on the other side.
---half of the y term squared goes in the y blank
and on the other side
Step 3
---factor the x trinomial
---factor the y trinomial
---Identify the conic

5. Ellipse
Problem:
Step 1 ---Rearrange the equation: ---x terms together on the left with a blank ---y terms together on the left with a blank constants on the right with 2 blanks Step 2 ---Fill in the blanks: ---half of the x term squared goes in the x blank and on the other side. ---half of the y term squared goes in the y blank and on the other side ---multiply them by the numbers outside for the other blanks Step 3 ---factor the x trinomial ---factor the y trinomial ---Divide both sides by the constant on the right hand side. ---Identify the conic

6. Parabola
Problem:
Step 1 ---non-squared variable is moved to the right along with the constant and a blank ---squared variable and corresponding terms on left with a blank Step 2 ---half the middle term squared goes in both blanks ---factor the squared trinomial ---factor out common value on the right Step 3 ---Move the factored out value to the other side by multiplying by the reciprocal. ---Identify the conic

7. Hyperbola
Problem:
Step 1 ---Rearrange the equation: ---x terms together on the left with a blank ---y terms together on the left with a blank ---constants on the right with 2 blanks Step 2 ---Fill in the blanks: ---half of the x term squared goes in the x blank and on the other side. ---half of the y term squared goes in the y blank and on the other side ---multiply them by the numbers outside for the other blanks Step 3 ---factor the x trinomial ---factor the y trinomial ---Divide both sides by the constant on the right hand side. ---Identify the conic

8. On Your Own a) b)

9. On Your Own c)
x² + 8y + 4x - 4 = 0 d)