To graph the parabola given by the equation [tex]\( y = -3x^2 - 18x - 24 \)[/tex] and find its key features, we need to determine its vertex, axis of symmetry, [tex]\( x \)[/tex]-intercepts, and [tex]\( y \)[/tex]-intercept. Let's go through this step by step:
### Step 1: Finding the Vertex
The vertex of a parabola given by the equation [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the vertex formula:
1. Calculate [tex]\( h \)[/tex]:
The [tex]\( h \)[/tex]-coordinate of the vertex is given by the formula:
[tex]\[
h = -\frac{b}{2a}
\][/tex]
For the equation [tex]\( y = -3x^2 - 18x - 24 \)[/tex], we have [tex]\( a = -3 \)[/tex] and [tex]\( b = -18 \)[/tex]:
[tex]\[
h = -\frac{-18}{2 \cdot -3} = \frac{18}{-6} = -3
\][/tex]
2. Calculate [tex]\( k \)[/tex]:
To find the [tex]\( k \)[/tex]-coordinate of the vertex, substitute [tex]\( h \)[/tex] into the original equation [tex]\( y = -3x^2 - 18x - 24 \)[/tex]:
[tex]\[
k = -3(-3)^2 - 18(-3) - 24
\][/tex]
Calculate each term:
[tex]\[
-3(-3)^2 = -3 \cdot 9 = -27
\][/tex]
[tex]\[
-18 \cdot -3 = 54
\][/tex]
Combining these, we get:
[tex]\[
k = -27 + 54 - 24 = 3
\][/tex]
So, the vertex is [tex]\((-3, 3)\)[/tex].
### Step 2: Finding the Axis of Symmetry
The axis of symmetry is the vertical line that passes through the vertex. Therefore, the axis of symmetry is:
[tex]\[
x = -3
\][/tex]
### Step 3: Finding the [tex]\( x \)[/tex]-Intercepts
The [tex]\( x \)[/tex]-intercepts are the points where the parabola crosses the [tex]\( x \)[/tex]-axis. To find the [tex]\( x \)[/tex]-intercepts, we set [tex]\( y = 0 \)[/tex] and solve the quadratic equation [tex]\( -3x^2 - 18x - 24 = 0 \)[/tex].
Using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
- [tex]\( a = -3 \)[/tex]
- [tex]\( b = -18 \)[/tex]
- [tex]\( c = -24 \)[/tex]
Calculate the discriminant:
[tex]\[
\Delta = b^2 - 4ac = (-18)^2 - 4(-3)(-24) = 324 - 288 = 36
\][/tex]
Since the discriminant is positive, we have two real solutions:
[tex]\[
x = \frac{-(-18) \pm \sqrt{36}}{2(-3)} = \frac{18 \pm 6}{-6}
\][/tex]
Calculate each root:
[tex]\[
x_1 = \frac{18 + 6}{-6} = \frac{24}{-6} = -4
\][/tex]
[tex]\[
x_2 = \frac{18 - 6}{-6} = \frac{12}{-6} = -2
\][/tex]
So, the [tex]\( x \)[/tex]-intercepts are [tex]\((-4, 0)\)[/tex] and [tex]\((-2, 0)\)[/tex].
### Step 4: Finding the [tex]\( y \)[/tex]-Intercept
The [tex]\( y \)[/tex]-intercept is the point where the parabola crosses the [tex]\( y \)[/tex]-axis. To find this, set [tex]\( x = 0 \)[/tex] in the equation [tex]\( y = -3x^2 - 18x - 24 \)[/tex]:
[tex]\[
y = -3(0)^2 - 18(0) - 24 = -24
\][/tex]
So, the [tex]\( y \)[/tex]-intercept is [tex]\((0, -24)\)[/tex].
### Summary
- Vertex: [tex]\((-3, 3)\)[/tex]
- Axis of Symmetry: [tex]\( x = -3 \)[/tex]
- [tex]\( x \)[/tex]-Intercepts: [tex]\((-4, 0)\)[/tex] and [tex]\((-2, 0)\)[/tex]
- [tex]\( y \)[/tex]-Intercept: [tex]\((0, -24)\)[/tex]
### Graphing the Parabola
To sketch the graph:
1. Plot the vertex [tex]\((-3, 3)\)[/tex].
2. Draw the axis of symmetry [tex]\( x = -3 \)[/tex].
3. Plot the [tex]\( x \)[/tex]-intercepts [tex]\((-4, 0)\)[/tex] and [tex]\((-2, 0)\)[/tex].
4. Plot the [tex]\( y \)[/tex]-intercept [tex]\((0, -24)\)[/tex].
5. Draw a smooth curve through these points, ensuring the parabola opens downwards (since [tex]\( a = -3 \)[/tex] is negative).
And the vertex for your solution is:
[tex]\[
\boxed{(-3, 3)}
\][/tex]
