Analyze each polynomial function and identify some of the key features of its graph. Select all that apply. 5. \( y=-3 x^{3}+21 x-18 \) \( \begin{array}{ll}\text { A. End Behavior: down, up } & \text { 6. } y=(x+3)(x-4) \\ \text { B. } y \text {-intercept }=-18 & \text { A. } \\ \begin{array}{ll}\text { C. } 2 \text { turning points } & \text { B. } \\ \text { C. } & 1 \text { turning point } \\ \text { D. function is even } & \text { D. } a \text { is negative }\end{array} \\ \text { E. Domain: }-\infty

Ask by Schultz Willis. in the United States

Jan 23,2025

For the polynomial function \( y = -3x^{3} + 21x - 18 \), let's break down some key features: 1. The end behavior of the graph, given the leading coefficient is negative and the degree is odd, is such that the graph will fall to the left (down) and rise to the right (up). So option A is correct. 2. The y-intercept can be found by evaluating the function at \( x = 0 \): \( y = -18 \). Thus, option B is correct. 3. The polynomial is cubic, which means it can have up to 2 turning points. So option C is correct. 4. Since the degree of the polynomial is odd, it is not an even function, making option D incorrect. 5. The domain of all polynomial functions is indeed all real numbers, so option E is correct. Now for the second function \( y = (x+3)(x-4) \): 1. To find the y-intercept, we calculate the function at \( x = 0 \): \( y = (0+3)(0-4) = -12 \), validating option E. 2. The function is a quadratic (degree 2) and opens upwards since it's in factored form and both factors are positive when multiplied. The fact that it opens upward implies that the coefficient on the leading term is positive, making option D incorrect (since the leading coefficient is not negative). 3. Quadratics have at most one turning point, located at the vertex of the parabola. Therefore, option C is correct, while option A’s implication of one turning point is true, but option B cannot be selected straight from the given polynomial representation as it is not clear without plotting. In summary, \( y = -3x^{3} + 21x - 18 \): A, B, C, E are true. For \( y = (x+3)(x-4) \): E is true.