How to find the axis of symmetry on a graph


Asked by admin @ in Mathematics viewed by 286 People


Find the axis of symmetry and the vertex of the graph of y = 3x2 + 2x.

The axis of symmetry is x=
and the vertex is

Answered by admin @



Step-by-step explanation:

Finding Vertex

Given

y\:=\:3x^2\:+\:2x

The vertex of an up-down facing parabola of the form

y=ax^2+bx+c\:\mathrm{is}\:x_v=-\frac{b}{2a}

The Parabola params are:

a=3,\:b=2,\:c=0

x_v=-\frac{b}{2a}

x_v=-\frac{2}{2\cdot \:3}

x_v=-\frac{1}{3}

\mathrm{Plug\:in}\:\:x_v=-\frac{1}{3}\:\mathrm{to\:find\:the}\:y_v\:\mathrm{value}

y_v=3\left(-\frac{1}{3}\right)^2+2\left(-\frac{1}{3}\right)

    =3\left(-\frac{1}{3}\right)^2-2\cdot \frac{1}{3}

    =\frac{1}{3}-\frac{2}{3}         ∵  3\left(-\frac{1}{3}\right)^2=\frac{1}{3}

    =\frac{1-2}{3}

    =\frac{-1}{3}

\mathrm{Apply\:the\:fraction\:rule}:\quad \frac{-a}{b}=-\frac{a}{b}

   =-\frac{1}{3}

y_v=-\frac{1}{3}

Therefore the parabola vertex is:

\left(-\frac{1}{3},\:-\frac{1}{3}\right)

\mathrm{If}\:a

\mathrm{If}\:a>0,\:\mathrm{then\:the\:vertex\:is\:a\:minimum\:value}

a=3

\mathrm{Minimum}\space\left(-\frac{1}{3},\:-\frac{1}{3}\right)

Finding symmetry

For a parabola in standard form y=ax^2+bx+c

the axis of symmetry is the vertical line that goes through the vertex x=\frac{-b}{2a}

\mathrm{Axis\:of\:Symmetry\:for}\:y=ax^2+bx+c\:\mathrm{is}\:x=\frac{-b}{2a}

a=3,\:b=2

x=\frac{-2}{2\cdot \:3}

x=-\frac{1}{3}

Therefore,

\mathrm{Axis\:of\:Symmetry\:for}\:y=3x^2+2x:\quad x=-\frac{1}{3}


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