Question

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graph the equation.
$y = -dfrac{3}{2}x^2 - 6x$

Explanation:

Step1: Rewrite the equation in vertex form

The given equation is \( y = -\frac{3}{2}x^2 - 6x \). First, factor out the coefficient of \( x^2 \) from the first two terms:
\( y = -\frac{3}{2}(x^2 + 4x) \)
To complete the square inside the parentheses, take half of the coefficient of \( x \) (which is \( 4 \)), square it (\( (\frac{4}{2})^2 = 4 \)), and add and subtract it inside the parentheses. But since we factored out \( -\frac{3}{2} \), we need to be careful with the constant term:
\( y = -\frac{3}{2}(x^2 + 4x + 4 - 4) \)
\( y = -\frac{3}{2}((x + 2)^2 - 4) \)
Distribute the \( -\frac{3}{2} \):
\( y = -\frac{3}{2}(x + 2)^2 + 6 \)
So the vertex form is \( y = -\frac{3}{2}(x + 2)^2 + 6 \), which means the vertex is at \( (-2, 6) \).

Step2: Find the x-intercepts

Set \( y = 0 \) in the original equation:
\( 0 = -\frac{3}{2}x^2 - 6x \)
Factor out \( -\frac{3}{2}x \):
\( 0 = -\frac{3}{2}x(x + 4) \)
This gives the solutions \( x = 0 \) and \( x = -4 \), so the x-intercepts are at \( (0, 0) \) and \( (-4, 0) \).

Step3: Analyze the graph

The coefficient of \( x^2 \) is \( -\frac{3}{2} \), which is negative, so the parabola opens downward. Wait, but the given graph in the image seems to open upward. There might be a mistake in the graph or in the analysis? Wait, no, let's recheck the original equation: \( y = -\frac{3}{2}x^2 - 6x \). Let's plug in \( x = -2 \), we get \( y = -\frac{3}{2}(4) - 6(-2) = -6 + 12 = 6 \), so vertex at \( (-2, 6) \). Let's plug in \( x = 0 \), \( y = 0 \); \( x = -4 \), \( y = -\frac{3}{2}(16) - 6(-4) = -24 + 24 = 0 \). Now, let's check the direction: since the coefficient of \( x^2 \) is negative, the parabola should open downward. But the graph in the image opens upward, which might be a different equation. However, to graph the given equation \( y = -\frac{3}{2}x^2 - 6x \), we can plot the vertex \( (-2, 6) \), the x-intercepts \( (0, 0) \) and \( (-4, 0) \), and then draw the parabola opening downward.

Answer:

To graph \( y = -\frac{3}{2}x^2 - 6x \):

1. Vertex: \( (-2, 6) \) (from vertex form \( y = -\frac{3}{2}(x + 2)^2 + 6 \))
2. x-intercepts: \( (0, 0) \) and \( (-4, 0) \)
3. Direction: Opens downward (since \( a = -\frac{3}{2} < 0 \))

Plot the vertex, x-intercepts, and draw the parabola opening downward. (Note: The graph in the provided image appears to open upward, which may be incorrect for this equation.)