Parabolas- Vertex, Axis of Symmetry & Direction

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Notes

\(\text{ Parabolas }\)
\(\underline{\text{Vertex Form}}\) \(\underline{\text{Standard Form}}\) \(\underline{\text{Intercepts Form}}\)
\(\text{Equation}\)
\(f(x)=a(x-h)^2+k\)
\(f(x)=ax^2+bx+c\)
\(f(x)=a(x-p)(x-q)\)
\(\text{Vertex}\)
\((h,k)\)
\(\left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)\)
\(\left(\frac{p+q}{2},f\left(\frac{p+q}{2}\right)\right)\)
\(\text{Axis of Symmetry}\)
\(x=h\)
\(x=\frac{-b}{2a}\)
\(x=\frac{p+q}{2}\)
\(\text{Opening Direction}\) \( \text{ opens up if }a\gt0,
\text{ opens down if } a\lt0\)


Problems

State the vertex, axis of symmetry and opening direction.

\(\textbf{1)}\) \(f(x)=3(x-4)^2+2\)
\(\textbf{2)}\) \(f(x)=-2(x+3)^2-1\)
\(\textbf{3)}\) \(f(x)=(x+5)^2-3\)
\(\textbf{4)}\) \(f(x)=4(x-4)^2\)
\(\textbf{5)}\) \(f(x)=-x^2+3\)
\(\textbf{6)}\) \(f(x)=2x^2-4x+5\)
\(\textbf{7)}\) \(f(x)=x^2-6x+3\)
\(\textbf{8)}\) \(f(x)=-x^2+4x+5\)
\(\textbf{9)}\) \(f(x)=4x^2+8x-2\)
\(\textbf{10)}\) \(f(x)=-2x^2+8x-3\)
\(\textbf{11)}\) \(f(x)=2(x-5)(x-1)\)
\(\textbf{12)}\) \(f(x)=-3(x+5)(x-3)\)
\(\textbf{13)}\) \(f(x)=5(x)(x-4)\)
\(\textbf{14)}\) \(f(x)=-(x+7)(x+5)\)
\(\textbf{15)}\) \(f(x)=4(x)(x-2)\)


See Related Pages\(\)

\(\bullet\text{ All Conic Section Notes}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Equation of a Circle}\)
\(\,\,\,\,\,\,\,\,(x-h)^2+(y-k)^2=r^2…\)
\(\bullet\text{ Parabolas}\)
\(\,\,\,\,\,\,\,\,y=a(x-h)^2+k…\)
\(\bullet\text{ Axis of Symmetry}\)
\(\,\,\,\,\,\,\,\,x=-\frac{b}{2a}…\)
\(\bullet\text{ Ellipses}\)
\(\,\,\,\,\,\,\,\,\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1…\)
\(\bullet\text{ Area of Ellipses}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\pi a b…\)
\(\bullet\text{ Hyperbolas}\)
\(\,\,\,\,\,\,\,\,\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1…\)
\(\bullet\text{ Conic Sections- Completing the Square}\)
\(\,\,\,\,\,\,\,\,x^2+8x+y^2−6y=11 \Rightarrow (x+4)^2+(y−3)^2=36…\)
\(\bullet\text{ Conic Sections- Parametric Equations}\)
\(\,\,\,\,\,\,\,\,x=h+r \cos{t}\)
\(\,\,\,\,\,\,\,\,y=k+r \sin{t}…\)
\(\bullet\text{ Degenerate Conics}\)
\(\,\,\,\,\,\,\,\,x^2−y^2=0…\)
\(\bullet\text{ Andymath Homepage}\)

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