Question

Graph $f\left(x\right)=-3x^2-6x-4$by using its properties.

Short answer

• The parabola opens downwards.
• The axis of symmetry is the line $x=-1$
• The vertex is $(-1,-1)$
• The y-intercept is $(0,-4)$
• Point symmetric to y-intercept $(-2,-4)$
• There is no x-intercepts.
• Graph of parabola :

Step-by-step solution

Step 1. Determine whether the parabola opens upward and downward. 

$$\begin{gathered} f\left(x\right)=a{x}^2+bx+c \\ f\left(x\right)=-3x^2-6x-4 \end{gathered}$$

Since a is -3 , the parabola opens downward.

Step 2. To find the equation of the axis of symmetry, use x=-b2a

$$\begin{gathered} x=-\frac{b}{2a} \\ x=-\frac{-6}{-3} \\ x=-2 \end{gathered}$$

The axis of symmetry is $x=-2$

The vertex is on the line $x=-2$

Step 3. Find the vertex. 

Find $f(-2)$

$$\begin{gathered} f\left(x\right)=-3x^2-6x-4 \\ f(-2)=-3{(-2)}^2-6(-2)-4 \\ f(-2)=-12+12-4 \\ f(-2)=-4 \end{gathered}$$

The vertex is$(-2,4)$

Find the y-intercept. Find the point symmetric to the y-intercept across the axis of symmetry.

The y-intercept occurs when x = 0 . Find f (0) .

$$\begin{gathered} f\left(x\right)=-3x^2-6x-4 \\ f\left(0\right)=-3{\left(0\right)}^2-6\left(0\right)-4 \\ f\left(0\right)=-4 \end{gathered}$$

The y-intercept is $(0,-4)$

The point $(0,-4)$is one unit to the right of the line of symmetry.

The point one unit to the left of the line of symmetry is $(-2,-4)$

Point symmetric to the y-intercept is$(-2,-4)$

Step 5. Find the x-intercepts.

The x-intercept occurs when f (x) = 0

Find f (x) = 0 .

$$\begin{gathered} f\left(x\right)=-3x^2-6x-4 \\ 0=-3x^2-6x-4 \end{gathered}$$

Test the discriminant.

$$\begin{gathered} b^2-4ac \\ {(-6)}^2-4(-3)(-4) \\ 36-48 \\ -12 \end{gathered}$$

Since the value of the discriminant is negative, there is no real solution and so no x-intercept.

Step 6. Graph the parabola   

Connect the points to graph the parabola. You may want to choose two more points for greater accuracy.