Q42. - Algebra 1

Question

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

$y=-x^2+10x-13$

Step-by-Step Solution

Answer

The vertex is $\left(5,12\right)$, the equation of the axis of symmetry is $x=5$ and the $y$-intercept is $-13$.

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; x 2 + 10 x &#8722; 13 .

Compare the quadratic function $y=-x^2+10x-13$ with the standard quadratic function $y=a{x}^2+bx+c$.

$a=-1,b=10,c=-13$

Substitute $a=-1$ and $b=10$ in $x=-\frac{b}{2a}$.

$$\begin{gathered} x=-\left(\frac{10}{2\left(-1\right)}\right) \\ x=-\left(\frac{10}{-2}\right) \\ x=-\left(-5\right) \\ x=5 \end{gathered}$$

So, the axis of symmetry is given by

$x=5$

Since, $a>0$.

So, the function has a minimum value at $x=5$.

Substitute $x=5$ in $y=-x^2+10x-13$.

$$\begin{gathered} y=-{\left(5\right)}^2+10\left(5\right)-13 \\ y=-25+50-13 \\ y=50-38 \\ y=12 \end{gathered}$$

So, the vertex point is located at the point $\left(5,12\right)$.

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

So, the $y$-intercept is $-13$.

 

Therefore the vertex is $\left(5,12\right)$, the equation of the axis of symmetry is $x=5$ and the $y$-intercept is $-13$.