f(x)=X^2-2X-3

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

The function declaration ${\displaystyle f\left(x\right)}$ varies according to ${\displaystyle x}$, but the input function ${\displaystyle X^2-2X-3}$ only contains the variable ${\displaystyle X}$. Assume ${\displaystyle f\left(X\right)=X^2-2X-3}$.

${\displaystyle f\left(X\right)=X^2-2X-3}$

Find the properties of the given parabola.

Rewrite the equation in vertex form.

Complete the square for ${\displaystyle X^2-2X-3}$.

Use the form ${\displaystyle a{X}^2+bX+c}$, to find the values of ${\displaystyle a}$, ${\displaystyle b}$, and ${\displaystyle c}$.

${\displaystyle a=1}$

${\displaystyle b=-2}$

${\displaystyle c=-3}$

Consider the vertex form of a parabola.

${\displaystyle a{(X+d)}^2+e}$

Find the value of ${\displaystyle d}$ using the formula ${\displaystyle d=\frac{b}{2a}}$.

${\displaystyle d=-1}$

Find the value of e${\displaystyle e}$ using the formula ${\displaystyle e=c-\frac{b^2}{4a}}$.

Substitute the values of ${\displaystyle c}$, ${\displaystyle b}$ and ${\displaystyle a}$ into the formula ${\displaystyle e=c-\frac{b^2}{4a}}$.

${\displaystyle e=-3-\frac{{(-2)}^2}{4\cdot 1}}$

Simplify the right side.

${\displaystyle e=-4}$

Substitute the values of ${\displaystyle a}$, ${\displaystyle d}$, and ${\displaystyle e}$ into the vertex form ${\displaystyle {(X-1)}^2-4}$.

${\displaystyle {(X-1)}^2-4}$

Set ${\displaystyle y}$ equal to the new right side.

${\displaystyle y={(X-1)}^2-4}$

Use the vertex form,${\displaystyle y=a{(X-h)}^2+k}$, to determine the values of ${\displaystyle a}$, ${\displaystyle h}$, and ${\displaystyle k}$.

${\displaystyle a=1}$

${\displaystyle h=1}$

${\displaystyle k=-4}$

Since the value of ${\displaystyle a}$ is positive, the parabola opens up.

Opens Up

Find the vertex ${\displaystyle (h,k)}$.

(1,−4)${\displaystyle (1,-4)}$

Find ${\displaystyle p}$, the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

${\displaystyle \frac{1}{4a}}$

Substitute the value of ${\displaystyle a}$ into the formula.

${\displaystyle \frac{1}{4\cdot 1}}$

Cancel the common factor of ${\displaystyle 1}$.

${\displaystyle \frac{1}{4}}$

Find the focus.

The focus of a parabola can be found by adding ${\displaystyle p}$ to the $y$-coordinate ${\displaystyle k}$ if the parabola opens up or down.

${\displaystyle (h,k+p)}$

Substitute the known values of ${\displaystyle h}$, ${\displaystyle p}$, and ${\displaystyle k}$ into the formula and simplify.

${\displaystyle (1,-\frac{15}{4})}$

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

${\displaystyle X=1}$

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting ${\displaystyle p}$ from the $y$-coordinate ${\displaystyle k}$ of the vertex if the parabola opens up or down.

${\displaystyle y=k-p}$

Substitute the known values of ${\displaystyle p}$ and ${\displaystyle k}$ into the formula and simplify.

${\displaystyle y=-\frac{17}{4}}$

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Up

Vertex: ${\displaystyle (1,-4)}$

Focus: ${\displaystyle (1,-\frac{15}{4})}$

Axis of Symmetry: ${\displaystyle X=1}$

Directrix: ${\displaystyle y=-\frac{17}{4}}$

Select a few ${\displaystyle X}$ values, and plug them into the equation to find the corresponding ${\displaystyle y}$ values. The ${\displaystyle X}$ values should be selected around the vertex.

${\displaystyle \begin{array}{cc}X& y\\ -1& 0\\ 0& -3\\ 1& -4\\ 2& -3\\ 3& 0\end{array}}$

Graph the parabola using its properties and the selected points.