Q41. - Algebra 1

Question

Find the vertex, the equation of the axis of symmetry, and the $y$-intercept of the given equation.

$y=-3x^2+6x-18$

Step-by-Step Solution

Answer

The vertex is $\left(1,-15\right)$, the equation of the axis of symmetry is $x=1$ and the $y$-intercept is $-18$.

Step 1. Define the standard form of the quadratic function.

A quadratic function, which is written in the form, $y=a{x}^2+bx+c$, where, $a\ne 0$ is called the standard form of the quadratic function.

Step 2. Define the vertex of the function y = a x 2 + b x + c .

For the function $y=a{x}^2+bx+c$

(1) If $a>0$, then the function has the minimum value at $x=-\frac{b}{2a}$ and the vertex is located at the minimum point.

(2) If $a<0$, then the function has the maximum value at $x=-\frac{b}{2a}$ and the vertex is located at the maximum point.

Step 3. Write the axis of symmetry for the function y = a x 2 + b x + c .

The axis of symmetry for the function $y=a{x}^2+bx+c$ is

$x=-\frac{b}{2a}$

Step 4. Define y -intercept of the function y = a x 2 + b x + c .

The $y$-intercept of the function $y=a{x}^2+bx+c$ is always at $c$.

Step 5. Calculate the vertex, the equation of the axis of symmetry, and the y -intercept of the function y = &#8722; 3 x 2 + 6 x &#8722; 18 .

Compare the quadratic function $y=-3x^2+6x-18$ with the standard quadratic function $y=a{x}^2+bx+c$.

$a=-3,b=6,c=-18$

Substitute $a=-3$ and $b=6$ in $x=-\frac{b}{2a}$.

$$\begin{gathered} x=-\left(\frac{6}{2\left(-3\right)}\right) \\ x=-\left(\frac{6}{-6}\right) \\ x=-\left(-1\right) \\ x=1 \end{gathered}$$

So, the axis of symmetry is given by 

$x=1$

Since, $a>0$.

So, the function has a minimum value at $x=1$.

Substitute $x=1$ in $y=-3x^2+6x-18$.

$$\begin{gathered} y=-3{\left(1\right)}^2+6\left(1\right)-18 \\ y=-3\left(1\right)+6-18 \\ y=-3+6-18 \\ y=-21+6 \\ y=-15 \end{gathered}$$

So, the vertex point is located at the point $\left(1,-15\right)$.

Since, the $y$-intercept is given by $c$.

So, the $y$-intercept is $-18$.

 

Therefore the vertex is $\left(1,-15\right)$, the equation of the axis of symmetry is $x=1$ and the $y$-intercept is $-18$.