Question

chapter 3.4: which of the following equations represents this function? 1 point. the image is a graph of a function. the options are: \\( f(x) = (x + 2)(x + 5) \\), \\( f(x) = -(x + 2)^2(x + 5) \\), \\( f(x) = (x - 2)^2(x - 5) \\), \\( f(x) = (x - 2)(x - 5) \\), \\( f(x) = -(x - 2)^2(x - 5) \\), \\( f(x) = -(x - 2)(x - 5) \\)

Explanation:

Step1: Identify x-intercepts

The graph crosses/touches the x-axis at $x=2$ (touches, so even multiplicity) and $x=5$ (crosses, so odd multiplicity). For a polynomial function, roots give factors: $(x-2)^2$ and $(x-5)$.

Step2: Determine leading coefficient sign

As $x\to+\infty$, $f(x)\to-\infty$. For a cubic polynomial (degree 3, odd), this means the leading coefficient is negative.

Step3: Match to options

Combine the factors and negative leading coefficient: $f(x)=-(x-2)^2(x-5)$. Eliminate other options:

• $f(x)=(x+2)(x+5)$: roots at $x=-2,-5$ (wrong intercepts)
• $f(x)=-(x+2)^2(x+5)$: roots at $x=-2,-5$ (wrong intercepts)
• $f(x)=(x-2)^2(x-5)$: leading coefficient positive (ends up as $x\to+\infty$)
• $f(x)=(x-2)(x-5)$: degree 2, parabola (wrong shape)
• $f(x)=-(x-2)(x-5)$: degree 2, parabola (wrong shape)

Answer:

f(x) = -(x - 2)²(x - 5)