I’ve Got a Fill-in

Learning Focus

Find missing parameters by moving between standard, vertex, and factored forms.

Big idea

• Standard: $y=ax^2+bx+c$ ($c$ is the $y$-intercept).
• Vertex: $y=a(x-h)^2+k$.
• Factored: $y=a(x-r)(x-s)$ (roots $r,s$).

Use a known point to solve for a missing parameter; expand or complete the square to switch forms.

Worked example

Find $a$ if $y=a(x-1)(x-5)$ passes through $(0,10)$:

$10=a(-1)(-5)=5a\implies a=2.$

Takeaways

• Substitute a known point to solve for one unknown.
• Expand to reach standard form; complete the square to reach vertex form.
• The vertex lies halfway between the roots.