Question

Graph $y=-x^2+5x-6$by using properties.

Short answer

The graph for the equation $y=-x^2+5x-6$is,

Step-by-step solution

Step 1 Given equation is,

$y=-x^2+5x-6$.

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

We get role="math" localid="1645268400289" $a=-1,b=5,c=-6$.

Since $a$is $-1$, the parabola opens downward.

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

Substitute the known values in the formula.

$$\begin{gathered} x=\frac{-5}{2\left(-1\right)} \\ =\frac{5}{2} \\ =2.5 \end{gathered}$$

Step 2 The vertex is on the line x=2.5.

Substitute $x=2.5$in the given equation.

$$\begin{gathered} y=-{\left(2.5\right)}^2+5\left(2.5\right)-6 \\ =-6.25+12.5-6 \\ =0.25 \end{gathered}$$

The vertex is, $\left(2.5,0.25\right)$.

The $y$intercept occurs when $x=0$.

So, substitute $x=0$in the given equation.

$$\begin{gathered} y=0+0-6 \\ =-6 \end{gathered}$$

The intercept is,$\left(0,-6\right)$.

Step 3 Now find the point symmetric to the y intercept.

The point $\left(0,-6\right)$is $2.5$units to the left of the line of symmetry. The point $2.5$units to the right of the line of symmetry is $\left(5,-6\right)$.

The point symmetric to the $y$-intercept is, $\left(5,-6\right)$.

The $x$-intercept occurs when $y=0$.

So, substitute $y=0$in the given equation.

$0=-x^2+5x-6$

Now factorize it.

role="math" localid="1645270321532" $$\begin{gathered} \left(x-2\right)\left(x-3\right)=0 \\ x=2,3 \end{gathered}$$

The$x$- intercepts are,$\left(2,0\right),\left(3,0\right)$.

Step 4 Now graph the given equation with the known values.