Question

deltamath back to home dml - polynomial graph key features due december 14 at 11:59 pm grade: 85% identify relative extrema graphically increasing or positive graphically end behavior graphically end behavior algebraically determine features of polynomial graph answer attempt 1 out of 3 the polynomial function f(x) is graphed below. fill in the form below regarding the features of this graph. the degree of f(x) is and the leading coefficient is . there are different real zeros and relative extremes.

Explanation:

Answer:

Step 1: Determine the degree of the polynomial

The end - behavior of a polynomial function is determined by the leading term \(a_nx^n\), where \(n\) is the degree and \(a_n\) is the leading coefficient. For the end - behavior, if the left - hand end and the right - hand end of the graph have the same direction (both going up or both going down), the degree \(n\) is even. If they have opposite directions, the degree \(n\) is odd. In the given graph, as \(x\to-\infty\) and \(x\to+\infty\), the ends of the graph go in the same direction (both going up or both going down? Wait, looking at the graph: the left - most part of the graph (as \(x\to-\infty\)): the upper curve goes to the left and up, the lower curve goes to the left and down? Wait, no, the graph has a shape. Wait, the number of turning points: the number of turning points of a polynomial function \(y = f(x)\) is at most \(n - 1\), where \(n\) is the degree of the polynomial. The graph shown has 2 turning points. So the degree \(n\) satisfies \(n-1\geq2\), so \(n\geq3\). Also, the end - behavior: let's check the leading coefficient sign. If the degree is odd, the ends go in opposite directions; if even, same. Wait, in the graph, when \(x\to+\infty\), the graph goes down, and when \(x\to-\infty\), the graph goes up. So the ends have opposite directions, so the degree is odd. The number of turning points is 2, so \(n-1 = 2\) (since the maximum number of turning points is \(n - 1\)), so \(n=3\)? Wait, no, if the number of turning points is 2, the degree is at least 3. But the end - behavior: opposite ends, so odd degree. Let's re - examine. The graph crosses the \(x\) - axis at \(x = 0\) (with a touch? Wait, no, the graph passes through the origin and has a turning point. Wait, the degree: the number of real zeros: the graph crosses the \(x\) - axis at \(x = 0\) (maybe a multiple root) and maybe another? Wait, no, the graph shown: let's count the number of \(x\) - intercepts. The graph crosses the \(x\) - axis at \(x = 0\) (but maybe with multiplicity). Wait, the degree: for a polynomial, the degree is related to the end - behavior and the number of turning points. The end - behavior: as \(x\to+\infty\), the graph goes down, as \(x\to-\infty\), the graph goes up. So the leading coefficient is negative and the degree is odd. The number of turning points: 2. So \(n-1=2\) implies \(n = 3\)? Wait, no, if \(n = 3\), the maximum number of turning points is 2. So degree \(n = 3\)? Wait, but the graph: let's see, the function \(f(x)\) has a graph that seems to have a root at \(x = 0\) (maybe with multiplicity 1) and two turning points. So the degree is 3 (cubic polynomial).

Step 2: Determine the leading coefficient sign

When the degree \(n\) is odd, if as \(x\to+\infty\), \(f(x)\to-\infty\), then the leading coefficient \(a_n\) is negative (because for \(y=a_nx^n\), when \(n\) is odd, if \(a_n>0\), as \(x\to+\infty\), \(y\to+\infty\); if \(a_n<0\), as \(x\to+\infty\), \(y\to-\infty\)). In our graph, as \(x\to+\infty\), the graph goes down ( \(f(x)\to-\infty\)), so the leading coefficient is negative.

Step 3: Determine the number of different real zeros

The graph crosses the \(x\) - axis at \(x = 0\) (and maybe another? Wait, no, the graph shown: it touches or crosses? Wait, the graph passes through the origin and has a turning point. Wait, the number of real zeros: the graph intersects the \(x\) - axis at \(x = 0\) (counting multiplicity? But the question says "different real zeros". Wait, the graph seems to cross the \(x\) - axis at \(x = 0\) (once) and maybe another? Wait, no, the graph as drawn: let's see, the curve comes from the top left, goes down, turns, goes up, passes through the origin, turns again, and goes down to the bottom right. Wait, so it crosses the \(x\) - axis at \(x = 0\) (one real zero) and maybe another? Wait, no, maybe I misread. Wait, the graph has a single \(x\) - intercept at \(x = 0\) (with multiplicity 3? No, for a cubic, if it has a root at \(x = 0\) with multiplicity 3, the graph would have a point of inflection at \(x = 0\). But here, there are two turning points. So maybe the cubic has one real zero (with multiplicity 1) and two complex zeros? No, the number of real zeros (counting multiplicity) of a cubic is 3 (by Fundamental Theorem of Algebra). But the number of different real zeros: let's see the graph. The graph crosses the \(x\) - axis at \(x = 0\) (once) and maybe another? Wait, no, the graph as shown: when we look at the \(x\) - axis, the graph passes through \(x = 0\) and maybe touches or crosses? Wait, the graph has a "wiggle" around the origin. Wait, maybe the number of different real zeros is 1 (since it only crosses the \(x\) - axis at \(x = 0\), even though it has a turning point there? No, if it has a turning point at \(x = 0\), the multiplicity is at least 2. Wait, this is getting confusing. Let's go back to the turning points. The number of turning points is 2, so degree is 3 (since \(n-1 = 2\)). The leading coefficient is negative (since as \(x\to+\infty\), \(f(x)\to-\infty\)). The number of different real zeros: the graph crosses the \(x\) - axis at \(x = 0\) (maybe with multiplicity 1) and since it's a cubic, there must be three real roots (counting multiplicity). But the number of different real zeros: if the graph only crosses the \(x\) - axis at \(x = 0\) (with multiplicity 3), but the graph has two turning points, which is consistent with a cubic with a triple root? No, a triple root at \(x = a\) would have a graph that touches the \(x\) - axis at \(x = a\) and has a point of inflection there, not two turning points. So maybe the cubic has one real zero (with multiplicity 1) and two complex conjugate zeros? But complex zeros come in pairs. Wait, no, the Fundamental Theorem of Algebra says that a cubic has three roots (real or complex, counting multiplicity). If it has one real root and two complex roots (which are conjugate), then the number of different real zeros is 1.

Step 4: Determine the number of relative extrema

The number of relative extrema (local maxima and minima) of a polynomial function is at most \(n - 1\), where \(n\) is the degree. For a cubic polynomial (\(n = 3\)), the maximum number of relative extrema is \(3-1=2\). Looking at the graph, we can see that there are 2 turning points (one local maximum and one local minimum), so the number of relative extrema is 2.

So, the degree of \(f(x)\) is \(3\) (odd), the leading coefficient is negative, there is \(1\) different real zero, and \(2\) relative extrema.