Question

consider the following sketch of a polynomial. what is the end behavior? as x→ -∞, y→ < and as x→ +∞, y→ < is this an odd or even degree polynomial? < is the polynomial positive or negative? < how many real zeros (x - intercepts) does it have? <

Explanation:

Step1: Count real - zeros

The graph crosses the x - axis at 3 points. So, the number of real zeros (x - intercepts) is 3.

Step2: Determine sign of leading coefficient

As \(x\to+\infty\), \(y\to+\infty\). For a polynomial \(y = a_nx^n+\cdots+a_0\), when \(n\) is odd and \(a_n> 0\), \(y\to+\infty\) as \(x\to+\infty\) and \(y\to-\infty\) as \(x\to-\infty\). So the polynomial is positive.

Step3: Determine degree

Since the graph has 2 turning points, and for a polynomial of degree \(n\), the number of turning points is at most \(n - 1\). So \(n-1=2\), then \(n = 3\) (an odd - degree polynomial).

Step4: Determine end - behavior

As \(x\to-\infty\), \(y\to-\infty\) and as \(x\to+\infty\), \(y\to+\infty\) because for an odd - degree polynomial with a positive leading coefficient, the end - behaviors are in opposite directions with \(y\) going to negative infinity as \(x\) goes to negative infinity and \(y\) going to positive infinity as \(x\) goes to positive infinity.

Answer:

1. 3
2. Positive
3. Odd
4. As \(x\to-\infty\), \(y\to-\infty\); as \(x\to+\infty\), \(y\to+\infty\)