Question

determine the equation for the cubic function shown below. you may leave your answer in factored form. the leading coefficient for the function is either 1 or -1.

Explanation:

Step1: Identify x-intercepts

From the graph, the cubic function crosses the x - axis at \(x = - 1\) (with multiplicity 1, since it crosses the axis) and touches or crosses at \(x = 2\)? Wait, no, looking at the graph, the roots (x - intercepts) are \(x=-1\) (crossing) and \(x = 2\) (crossing? Wait, no, the graph has a root at \(x=-1\) (where it crosses the x - axis) and a root at \(x = 2\) (crossing), and also, since it's a cubic, there should be three real roots. Wait, the graph touches the x - axis at \(x = 0\)? Wait, no, the graph has a local minimum at \(y = 0\) (wait, the graph passes through \(x=-1\), touches at \(x = 0\)? Wait, no, let's re - examine. The graph crosses the x - axis at \(x=-1\) (when \(x=-1\), \(y = 0\)), touches the x - axis at \(x = 0\) (a repeated root, multiplicity 2) and crosses at \(x = 2\)? Wait, no, the leading coefficient is \(\pm1\). Let's recall that a cubic function in factored form is \(y=a(x - r_1)(x - r_2)(x - r_3)\), where \(r_1,r_2,r_3\) are the roots.

Looking at the graph: when \(x=-1\), \(y = 0\); when \(x = 0\), the graph touches the x - axis (so \(x = 0\) is a root with multiplicity 2), and when \(x = 2\), \(y=0\). Wait, no, the graph: let's check the behavior. As \(x\to-\infty\), \(y\to+\infty\) and as \(x\to+\infty\), \(y\to-\infty\), so the leading coefficient \(a=-1\) (since for a cubic \(y = ax^3+\cdots\), if \(a\lt0\), as \(x\to+\infty\), \(y\to-\infty\) and \(x\to-\infty\), \(y\to+\infty\)).

Now, the roots: the graph crosses the x - axis at \(x=-1\), touches the x - axis at \(x = 0\) (so \(x = 0\) is a double root) and crosses at \(x = 2\)? Wait, no, when \(x = 0\), the graph has a local minimum at \(y = 0\)? Wait, the graph at \(x = 0\) is on the x - axis (the point \((0,0)\) is on the graph). Then, the roots are \(x=-1\), \(x = 0\) (with multiplicity 2), and \(x = 2\)? Wait, no, a cubic has three roots (counting multiplicities). Let's see: the graph passes through \((-1,0)\), \((0,0)\) (touching, so multiplicity 2), and \((2,0)\)? Wait, no, when \(x = 2\), \(y = 0\). Wait, let's write the factored form. If the roots are \(x=-1\), \(x = 0\) (multiplicity 2), and \(x = 2\), then the factored form is \(y=a(x + 1)x^{2}(x - 2)\). But the leading coefficient \(a\) is either \(1\) or \(-1\). Let's check the end - behavior. For \(y=a(x + 1)x^{2}(x - 2)=a(x^{4}-x^{3}-2x^{2})\), but wait, that's a quartic. Oh, I made a mistake. It's a cubic, so three roots. So the roots are \(x=-1\), \(x = 0\), and \(x = 2\)? Wait, no, the graph: when \(x=-1\), \(y = 0\); when \(x = 0\), \(y = 0\) (touching, so multiplicity 2), but that would be a quartic. Wait, no, cubic: degree 3. So the roots are \(x=-1\), \(x = 0\), and \(x = 2\) with multiplicities such that the sum of multiplicities is 3. So one root with multiplicity 2 and one with multiplicity 1. Let's assume the roots are \(x = 0\) (multiplicity 2) and \(x = 2\) (multiplicity 1), and \(x=-1\) (multiplicity 0)? No, that's wrong. Wait, let's look at the graph again. The graph crosses the x - axis at \(x=-1\) (so \(x=-1\) is a root, multiplicity 1), touches the x - axis at \(x = 0\) (multiplicity 2), so the cubic function is \(y=a(x + 1)x^{2}\). But wait, that's a cubic? \(y=a(x + 1)x^{2}=a(x^{3}+x^{2})\). Let's check the leading coefficient. As \(x\to+\infty\), if \(a=-1\), then \(y=-x^{3}-x^{2}\), as \(x\to+\infty\), \(y\to-\infty\) (correct), as \(x\to-\infty\), \(y\to+\infty\) (correct). Now, let's check the value at \(x = 2\). If \(y=-x^{2}(x + 1)\), when \(x = 2\), \(y=-4\times(3)=-12\), which matches the graph (at \(x = 2\), \(y = 0\)? No, wait, I made a mistake. The graph at \(x = 2\) is \(y = 0\). So my earlier analysis is wrong.

Wait, the correct roots: the graph crosses the x - axis at \(x=-1\), \(x = 0\), and \(x = 2\)? No, when \(x = 0\), the graph is at \(y = 0\) (touching), \(x=-1\) at \(y = 0\) (crossing), and \(x = 2\) at \(y = 0\) (crossing). So the factored form is \(y=a(x + 1)x(x - 2)\). Now, let's find \(a\). The leading coefficient \(a\) is \(\pm1\). Let's check the end - behavior. For \(y=a(x + 1)x(x - 2)=a(x^{3}-x^{2}-2x)\). As \(x\to+\infty\), if \(a=-1\), then \(y=-x^{3}+x^{2}+2x\), as \(x\to+\infty\), \(y\to-\infty\) (correct), as \(x\to-\infty\), \(y\to+\infty\) (correct). Now, let's check a point on the graph. Let's take \(x = 1\). From the graph, at \(x = 1\), \(y = 1\) (approx). Let's plug \(x = 1\) into \(y=-x(x + 1)(x - 2)\): \(y=-1\times(2)\times(-1)=2\). Not matching. Wait, maybe the roots are \(x=-1\), \(x = 0\) (multiplicity 2), and \(x = 2\) is not a root. Wait, the graph: when \(x = 2\), \(y\) is negative, going to \(-\infty\). Wait, let's start over.

The general form of a cubic function is \(y=a(x - r_1)(x - r_2)(x - r_3)\). The graph has a root at \(x=-1\) (crossing, so \(r_1=-1\)), a root at \(x = 0\) (touching, so \(r_2 = r_3=0\), multiplicity 2), so \(y=a(x + 1)x^{2}\). Now, let's check the leading coefficient. As \(x\to+\infty\), if \(a=-1\), \(y=-x^{3}-x^{2}\), as \(x\to+\infty\), \(y\to-\infty\) (correct), as \(x\to-\infty\), \(y\to+\infty\) (correct). Now, let's check the value at \(x = 1\). From the graph, at \(x = 1\), \(y = 1\) (approx). Plugging \(x = 1\) into \(y=-x^{2}(x + 1)\): \(y=-1\times(2)=-2\). Not matching. Wait, maybe the roots are \(x=-1\), \(x = 0\), and \(x = 2\) with \(a=-1\). So \(y=-1(x + 1)x(x - 2)\). Let's expand this: \(y=-(x^{3}-2x^{2}+x^{2}-2x)=-(x^{3}-x^{2}-2x)=-x^{3}+x^{2}+2x\). Now, check \(x = 1\): \(y=-1 + 1+2=2\). The graph at \(x = 1\) has a local maximum, and the \(y\) - value is around 1 - 2, which is close. Now, check \(x=-1\): \(y=-(-1)^{3}+(-1)^{2}+2\times(-1)=1 + 1-2=0\) (correct). \(x = 0\): \(y=0+0 + 0=0\) (correct). \(x = 2\): \(y=-8 + 4+4=0\) (correct). And the leading coefficient is \(-1\), which matches the given condition (leading coefficient is either 1 or - 1).

Step2: Write the factored form

Since the roots are \(x=-1\), \(x = 0\), and \(x = 2\), and the leading coefficient \(a=-1\), the factored form is \(y=-x(x + 1)(x - 2)\).

Answer:

\(y=-x(x + 1)(x - 2)\)