Question

graph $g(x) = -3x^2 - 12x - 14$.

Explanation:

Step1: Rewrite in vertex form

To graph the quadratic function \( g(x) = -3x^2 - 12x - 14 \), we first convert it to vertex form \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex.
Factor out the coefficient of \( x^2 \) from the first two terms:
\( g(x) = -3(x^2 + 4x) - 14 \)
Complete the square inside the parentheses. For \( x^2 + 4x \), we take half of 4 (which is 2), square it (which is 4), and add and subtract it inside the parentheses. But since there's a factor of -3 outside, we have to be careful:
\( g(x) = -3(x^2 + 4x + 4 - 4) - 14 \)
\( g(x) = -3((x + 2)^2 - 4) - 14 \)
Distribute the -3:
\( g(x) = -3(x + 2)^2 + 12 - 14 \)
\( g(x) = -3(x + 2)^2 - 2 \)

Step2: Identify vertex and direction

From the vertex form \( g(x) = -3(x + 2)^2 - 2 \), the vertex \((h, k)\) is \((-2, -2)\). Since the coefficient of \((x + 2)^2\) is -3 (which is negative), the parabola opens downward.

Step3: Find additional points (optional but helpful)

Let's find the y-intercept by setting \( x = 0 \):
\( g(0) = -3(0)^2 - 12(0) - 14 = -14 \), so the y-intercept is \((0, -14)\).
Find the x-intercepts by setting \( g(x) = 0 \):
\( -3x^2 - 12x - 14 = 0 \)
Multiply both sides by -1: \( 3x^2 + 12x + 14 = 0 \)
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 3 \), \( b = 12 \), \( c = 14 \):
Discriminant \( D = 12^2 - 4(3)(14) = 144 - 168 = -24 \)
Since the discriminant is negative, there are no real x-intercepts.

Step4: Plot the vertex and other points

Plot the vertex \((-2, -2)\). Since the parabola opens downward, we can find a few more points. For example, when \( x = -1 \):
\( g(-1) = -3(-1)^2 - 12(-1) - 14 = -3 + 12 - 14 = -5 \), so the point is \((-1, -5)\)
When \( x = -3 \):
\( g(-3) = -3(-3)^2 - 12(-3) - 14 = -27 + 36 - 14 = -5 \), so the point is \((-3, -5)\)
When \( x = 1 \):
\( g(1) = -3(1)^2 - 12(1) - 14 = -3 - 12 - 14 = -29 \), so the point is \((1, -29)\) (but this is far down, maybe not necessary for a rough graph)

Now, using the vertex \((-2, -2)\), the y-intercept \((0, -14)\), and the symmetric points \((-1, -5)\) and \((-3, -5)\), we can sketch the parabola opening downward with vertex at \((-2, -2)\).

Answer:

To graph \( g(x) = -3x^2 - 12x - 14 \):

1. Vertex: \((-2, -2)\) (from vertex form \( g(x) = -3(x + 2)^2 - 2 \)).
2. Direction: Opens downward (since the coefficient of \( x^2 \) is negative).
3. Key Points: Vertex \((-2, -2)\), y-intercept \((0, -14)\), symmetric points \((-1, -5)\) and \((-3, -5)\).

Plot these points and draw a smooth downward - opening parabola through them.