Question

what degree is this polynomial function? a second b third c fourth d fifth

Explanation:

Step1: Analyze the graph's turning points

A polynomial of degree \( n \) has at most \( n - 1 \) turning points (local maxima or minima). The given graph has 3 turning points (2 local maxima/minima? Wait, looking at the graph: let's count the number of times the graph changes direction. From the left, it comes up, has a peak, then a valley, then a peak? Wait, no, let's see: the graph has 3 turning points? Wait, no, let's check again. Wait, the graph: let's count the number of critical points (where the slope is zero). The graph has 3 turning points? Wait, no, the number of turning points is related to the degree. For a polynomial, the number of turning points is at most \( \text{degree} - 1 \). So if the graph has 3 turning points, then the degree is at least \( 3 + 1 = 4 \)? Wait, no, wait: let's look at the end behavior. The leading term's degree determines the end behavior. As \( x \to \infty \), the graph goes up, and as \( x \to -\infty \), the graph goes down? Wait, no, looking at the graph: when \( x \) is large positive, the graph is increasing (going up), and when \( x \) is large negative, the graph is decreasing (going down). Wait, no, the left end: as \( x \to -\infty \), the graph goes down (since it's coming from the bottom left), and as \( x \to \infty \), it goes up. So the leading coefficient is positive, and the degree is odd? Wait, no, wait: if the end behavior is opposite (one end up, one end down), the degree is odd. But the number of turning points: let's count. The graph has 3 turning points? Wait, no, let's see the graph: it has a local maximum, a local minimum, and another local maximum? Wait, no, the graph as shown: let's count the number of times the direction changes. Let's see: from the left (x very negative), the graph is decreasing (going down), then it has a local maximum (turns up), then a local minimum (turns down), then a local maximum (turns up)? Wait, no, the graph in the image: let's see the x-axis from -10 to 10, y-axis from -10 to 10. The graph crosses the x-axis, has a peak, then a valley, then a peak? Wait, no, the graph has 3 turning points? Wait, no, the number of turning points is 3? Wait, no, the formula is: number of turning points \( \leq \text{degree} - 1 \). So if there are 3 turning points, degree \( \geq 4 \). But also, the end behavior: as \( x \to \infty \), \( y \to \infty \); as \( x \to -\infty \), \( y \to -\infty \)? Wait, no, when \( x \) is very negative, the graph is going down (since it's coming from below), and when \( x \) is very positive, it's going up. So the leading term is of odd degree? But wait, if the degree is 4, the end behavior would be both ends up or both ends down (even degree). But here, one end is up, one end is down (odd degree). Wait, that's a contradiction. Wait, maybe I misread the end behavior. Wait, looking at the graph: when \( x \) approaches \( +\infty \), the graph is increasing (going up), and when \( x \) approaches \( -\infty \), the graph is decreasing (going down). So that's odd degree. But the number of turning points: if it's odd degree, the number of turning points is even? Wait, no, odd degree polynomials have an even number of turning points? Wait, no: a cubic (degree 3) has at most 2 turning points (even number). A quartic (degree 4) has at most 3 turning points (odd number). Wait, the graph in the image: let's count the turning points. Let's see: the graph has 3 turning points (local max, local min, local max? Or local min, local max, local min?). Wait, the graph as shown: let's see the number of times it changes direction. Let's count: 1. First, it comes from the bottom left (x very negative, y very negative), then it rises to a local maximum (turns down), then falls to a local minimum (turns up), then rises to a local maximum (turns down)? No, wait, the graph in the image: looking at the x-axis, around x=-2, x=0, etc. Wait, maybe the graph has 3 turning points, so the degree is 4 (since 3 turning points, degree is 4, because degree -1 = 3 turning points, so degree 4). But wait, the end behavior: if degree is 4 (even), both ends should go in the same direction. But in the graph, as x→∞, it goes up, and as x→-∞, it goes down? Wait, no, maybe I misread the end behavior. Wait, the left end: when x is very negative, the graph is going down (y→-∞), and when x is very positive, y→+∞. So that's odd degree. But a cubic (degree 3) has at most 2 turning points. The graph has 3 turning points, so degree must be at least 4. Wait, maybe the graph has 3 turning points, so degree is 4 (quartic). Wait, the options are second (degree 2, parabola, 1 turning point), third (degree 3, 2 turning points), fourth (degree 4, 3 turning points), fifth (degree 5, 4 turning points). The graph has 3 turning points, so degree is 4 (fourth). So the correct option is C.

Step2: Confirm with end behavior and turning points

- End behavior: For even degree, both ends go same direction; for odd, opposite. Wait, but in the graph, as x→∞, y→∞, and as x→-∞, y→-∞? Wait, no, maybe the left end is actually going up? Wait, maybe I made a mistake. Wait, the graph: when x is very negative, the graph is coming from the bottom (y negative) and going up, then has a peak, then a valley, then a peak? Wait, no, the graph in the image: let's look again. The graph crosses the x-axis, has a local maximum, then a local minimum, then a local maximum? Wait, no, the number of turning points: let's count the number of times the slope changes sign. The graph has 3 turning points (3 times the direction changes). So number of turning points = 3. Then degree = number of turning points + 1 = 4 (since degree -1 ≥ number of turning points). So degree is 4, which is fourth degree. So the answer is C.

Answer:

C. fourth