Q54. - Algebra 1

Question

Graph the function.

$y=-2x^2-8x+2$

Step-by-Step Solution

Answer

The graph of the function $y=-2x^2-8x+2$ 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 = &#8722; 2 x 2 &#8722; 8 x + 2 .

Compare the quadratic function $y=-2x^2-8x+2$ with the standard quadratic function $y=a{x}^2+bx+c$.

$a=-2,b=-8,c=2$

Substitute $a=-2$ and $b=-8$ in $x=-\frac{b}{2a}$.

$$\begin{gathered} x=-\left(\frac{-8}{2\left(-2\right)}\right) \\ x=-\left(\frac{-8}{-4}\right) \\ x=-\left(2\right) \\ x=-2 \end{gathered}$$

Since, $a<0$.

So, the graph opens downward and has a maximum value at $x=-2$.

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

So, the $y$-intercept is 2. 

The graph of the function $y=-2x^2-8x+2$ is shown below.