Question

the graph of the function below, including correct signs, x-intercepts and y-intercepts
f(x) = -x²(2x + 12)(x + 4)
plot the y-intercept and the roots. click on the graph to plot a point. click a point again to delete i
done plotting

Explanation:

Step1: Find x-intercepts (roots)

To find the x - intercepts, set \( f(x)=0 \). So, we solve the equation \( -x^{2}(2x + 12)(x + 4)=0 \).
Using the zero - product property, if \( abcd = 0 \), then at least one of \( a,b,c,d \) is zero.

• For \( -x^{2}=0 \), we have \( x = 0 \) (with multiplicity 2).
• For \( 2x+12 = 0 \), we solve for \( x \): \( 2x=-12\Rightarrow x=-6 \).
• For \( x + 4=0 \), we solve for \( x \): \( x=-4 \).

Step2: Find y - intercept

To find the y - intercept, set \( x = 0 \) in the function \( f(x) \).
Substitute \( x = 0 \) into \( f(x)=-x^{2}(2x + 12)(x + 4) \).
\( f(0)=-(0)^{2}(2(0)+12)(0 + 4)=0 \). Wait, let's expand the function first:
\( f(x)=-x^{2}(2x + 12)(x + 4)=-x^{2}(2x^{2}+8x+12x + 48)=-x^{2}(2x^{2}+20x + 48)=-2x^{4}-20x^{3}-48x^{2} \)
When \( x = 0 \), \( f(0)=0 \). Wait, maybe we made a mistake. Let's recalculate the y - intercept by plugging \( x = 0 \) into the original factored form:
\( f(0)=-0^{2}(2(0)+12)(0 + 4)=0 \). Wait, that can't be right. Wait, let's expand the original function correctly:
\( (2x + 12)(x + 4)=2x(x + 4)+12(x + 4)=2x^{2}+8x+12x + 48=2x^{2}+20x + 48 \)
Then \( f(x)=-x^{2}(2x^{2}+20x + 48)=-2x^{4}-20x^{3}-48x^{2} \)
When \( x = 0 \), \( f(0)=0 \). But let's check again. Wait, maybe there is a miscalculation. Wait, no, if \( x = 0 \), the first factor \( -x^{2}=0 \), so the whole function is zero. So the y - intercept is at \( (0,0) \), which is also an x - intercept. But let's verify by expanding the function completely:
\( f(x)=-x^{2}(2x + 12)(x + 4)=-x^{2}(2x(x + 4)+12(x + 4))=-x^{2}(2x^{2}+8x+12x + 48)=-x^{2}(2x^{2}+20x + 48)=-2x^{4}-20x^{3}-48x^{2} \)
When \( x = 0 \), \( y = 0 \).

So the x - intercepts (roots) are at \( x=-6 \), \( x = - 4 \), and \( x = 0 \) (with multiplicity 2), and the y - intercept is at \( (0,0) \).

Answer:

The x - intercepts (roots) are at \( x=-6 \), \( x=-4 \), and \( x = 0 \) (the point \( (0,0) \) is both an x - intercept and a y - intercept). To plot the points:

• For \( x=-6 \), the point is \( (-6,0) \).
• For \( x=-4 \), the point is \( (-4,0) \).
• For \( x = 0 \), the point is \( (0,0) \) (which is also the y - intercept).