Question

Graph$x=-y^2+2y-3$by using properties.

Short answer

The graph for the function $x=-y^2+2y-3$is,

Step-by-step solution

Step 1 Given equations are,

$x=-y^2+2y-3$.

Compare the given equation with $x=a{y}^2+by+c$.

Identify the constants $a,b,c$.

$a=-1,b=2,c=-3$

Since $a=-1$, the parabola opens leftward.

To find the axis of symmetry, $y=\frac{-b}{2a}$.

Substitute the known values in the formula.

$$\begin{gathered} y=\frac{-2}{2\left(-1\right)} \\ =1 \end{gathered}$$

Step 2 The vertex is on the line y=1.

Substitute $y=1$in the given equation.

$$\begin{gathered} x=-{\left(1\right)}^2+2\left(1\right)-3 \\ =-1+2-3 \\ =-2 \end{gathered}$$

The vertex is, $\left(-2,1\right)$.

To find the $y$- intercepts, substitute $x=0$in the given equation.

$0=-y^2+2y-3$

The solution of this quadratic equation are the complex numbers. So, there are no $y$- intercepts.

To find $x$- intercepts, substitute $y=0$in the given equation.

$$\begin{gathered} x=0+0-3 \\ x=-3 \end{gathered}$$

The $x$- intercepts are, $\left(-3,0\right)$.

The point $\left(-3,0\right)$is one unit below the line of symmetry. The line of symmetry one unit above the line of symmetry is, $\left(-3,2\right)$.

Step 3 Now plot a graph by using the known values.