Math · 25 · Advanced Math

Quadratic Graphs

Notes

$a>0$ opens up (minimum); $a<0$ opens down (maximum).
Vertex form $y=a(x-h)^2+k$ : vertex $(h,k)$ ; the sign flips on $h$ only.
Factored form $y=a(x-p)(x-q)$ : x-intercepts $p$ and $q$ .
Standard form $y=ax^2+bx+c$ : y-intercept is $(0,c)$ .
Axis of symmetry runs through the vertex, halfway between the x-intercepts.
Larger $|a|$ = narrower parabola.
“Which form displays…” questions: pick the form where that feature is a visible constant.

Vertex form

y=a(x-h)^2+k

Vertex (h, k)

Factored form

y=a(x-p)(x-q)

x-intercepts p and q

Axis of symmetry

x=-\dfrac{b}{2a}=\dfrac{p+q}{2}

Vertex y-value

k=f(h)

Plug the axis x-value back in

Symmetry

f(h+d)=f(h-d)

Equal heights either side of the axis

Graph it in Desmos and click the vertex and intercepts. Save algebra for “which form shows” questions.

10 official SAT questions on Quadratic Graphs, easiest first.