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Vertex Form $f\left(x\right)=a\left(x-h\right)^2+k$

Graph the following functions on the grid above.

$f\left(x\right)=\left(x+1\right)^2+3$

$g\left(x\right)=2\left(x+1\right)^2+3$

$h\left(x\right)=\frac{1}{3}\left(x+1\right)^2+3$

$j\left(x\right)=-\left(x+1\right)^2+3$

All of the functions have the form $a\left(x+1\right)^2+3$.
The coordinates of the vertex is and
the equation of the axis of symmetry is $x=$

Graph the following functions on the grid above.

$t\left(x\right)=3\left(x+4\right)^2+2$

$u\left(x\right)=\frac{1}{2}\left(x+3\right)^2-5$

$v\left(x\right)=-1.5\left(x-2\right)^2+4$

$w\left(x\right)=-\frac{1}{4}\left(x-6\right)^2-1$

Write down the coordinates of the vertex and equation fo the axis of symmetry for each function.

vertex:

axis of symmetry:

Now we will convert $f\left(x\right)=2x^2+12x+14$ to the form $f\left(x\right)=a\left(x-h\right)^2+k$. Fill in the blanks.

$f\left(x\right)=2x^2+12x+14$

$f\left(x\right)=$ $\left(x^2+6x+7\right)$

(take out a factor of $\square$)

$f\left(x\right)=2\bigl(x^2+6x\phantom{AAA}+7\phantom{AAAI}\bigl)$

(move $+7$ to the right)

$f\left(x\right)=2\bigl(x^2+6x+$ $+7-$ $\bigl)$

(add and subtract)

$\phantom{f\left(x\right)=2\bigl(}\underbrace{\phantom{.............}}_{\phantom{AAAAAAA}}$

$f\left(x\right)=2\bigl\{\bigl(x+$ $\bigl)^2-$ $\bigl\}$

(complete the square)

$f\left(x\right)=2\bigl(x+3\bigl)^2-$

(distribute $2$ to inside the $\bigl\{ \bigl\}$ brackets)

Write $f\left(x\right)=3x^2-12x+2$ in the form $f\left(x\right)=a\left(x-h\right)^2+k$. Fill in the blanks.

$f\left(x\right)=3x^2-12x+2$

$f\left(x\right)=$ $\left(x^2-4x+\frac{2}{3}\right)$

(take out a factor of $\square$)

$f\left(x\right)=3\bigl(x^2-4x\phantom{AAA}+\frac{2}{3}\phantom{AAAI}\bigl)$

(move $+\frac{2}{3}$ to the right)

$f\left(x\right)=3\bigl(x^2-4x+$ $+\frac{2}{3}-$ $\bigl)$

(add and subtract)

$\phantom{f\left(x\right)=2\bigl(}\underbrace{\phantom{.............}}_{\phantom{AAAAAAA}}$

$f\left(x\right)=3\bigl\{\bigl(x-$ $\bigl)^2-$ $\bigl\}$

(complete the square) *use improper fractions

$f\left(x\right)=3\bigl(x-2\bigl)^2-$

(distribute $3$ to inside the $\bigl\{ \bigl\}$ brackets)

Write $f\left(x\right)=5x^2-15x-7$ in the form $f\left(x\right)=a\left(x-h\right)^2+k$. Fill in the blanks.

$f\left(x\right)=5x^2-15x-7$

$f\left(x\right)=$ $\bigl(x^2-3x-$ $\bigl)$

(take out a factor of $\square$) *use improper fractions

$f\left(x\right)=5\bigl(x^2-3x\phantom{AAA}-\frac{7}{5}\phantom{AAAI}\bigl)$

(move $-\frac{7}{5}$ to the right)

$f\left(x\right)=5\bigl(x^2-3x+$ $-\frac{7}{5}-$ $\bigl)$

(add and subtract)

$\phantom{f\left(x\right)=2\bigl(}\underbrace{\phantom{.............}}_{\phantom{AAAAAAA}}$

$f\left(x\right)=5\bigl\{\bigl(x-$ $\bigl)^2-$ $\bigl\}$

(complete the square) *use improper fractions

$f\left(x\right)=5\bigl(x-\frac{3}{2}\bigl)^2-$

(distribute $5$ to inside the $\bigl\{ \bigl\}$ brackets)

Write $f\left(x\right)=-2x^2-10x+5$ in the form $f\left(x\right)=a\left(x-h\right)^2+k$. Fill in the blanks.

$f\left(x\right)=-2x^2-10x+5$

$f\left(x\right)=$ $\bigl(x^2+5x-$ $\bigl)$

(take out a factor of $\square$) *use improper fractions

$f\left(x\right)=-2\bigl(x^2+5x\phantom{AAA}-\frac{5}{2}\phantom{AAAI}\bigl)$

(move $-\frac{5}{2}$ to the right)

$f\left(x\right)=-2\bigl(x^2+5x+$ $-\frac{5}{2}-$ $\bigl)$

(add and subtract)

$\phantom{f\left(x\right)=-2\bigl(}\underbrace{\phantom{.............}}_{\phantom{AAAAAAA}}$

$f\left(x\right)=-2\bigl\{\bigl(x+$ $\bigl)^2-$ $\bigl\}$

(complete the square) *use improper fractions

$f\left(x\right)=-2\bigl(x+\frac{5}{2}\bigl)^2+$

(distribute $-2$ to inside the $\bigl\{ \bigl\}$ brackets)

Practice

Write the following in the form $a\left(x-h\right)^2+k$. Give your answers using improper fractions.

$2x^2+8x+5$

$2x^2-8x+3$

$2x^2+6x+5$

$2x^2-18x-3$

$3x^2+12x-4$

$5x^2-10x+2$

$3x^2+9x+5$

$4x^2-12x-1$

$-x^2-2x+3$

$-x^2+6x-2$

$-x^2-5x-1$

$-x^2+5x-3$

$-3x^2-21x-1$

$-3x^2+6x+5$