Q55. - Algebra 1

Question

Graph the function.

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

Step-by-Step Solution

Answer

The graph of the function $y=x^2+6x-6$ is

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 maximum or minimum point of the function y = a x 2 + b x + c .

The graph of the function $y=a{x}^2+bx+c$,

Opens upward and has a minimum value at $x=-\frac{b}{2a}$, when $a>0$.

Opens downward and has a maximum value at $x=-\frac{b}{2a}$,  when $a<0$. 

 

Step 3. 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 4. Plot the graph of the function y = x 2 + 6 x &#8722; 6 .

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

$a=1,b=6,c=-6$

Substitute $a=1$ and $b=6$ in $x=-\frac{b}{2a}$.

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

Since, $a>0$.

So, the graph opens upward and has a minimum value at $x=-3$.

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

So, the $y$-intercept is $-6$. 

The graph of the function $y=x^2+6x-6$ is shown below.