Question

Esmeralda and Heinz both work in a science lab. In order to secure funding for their future experiments, they must present their findings to some investors. The investors are not interested in listening to formulas. They want to see graphs because they are visual people. Unfortunately, Esmeralda and Heinz are having some difficulties.
Esmeralda and Heinz are working to graph a polynomial function, f(x). Esmeralda says that the third-degree polynomial has four intercepts. Heinz argues that the function only crosses the x-axis three times. Is there a way for them both to be correct? Explain your answer.

Heinz has a list of possible functions. Pick one of the g(x) functions below, show how to find the zeros, and then describe to Heinz the other key features of g(x).
g(x) = x3 – x2 – 4x + 4
g(x) = x3 + 2x2 – 9x – 18
g(x) = x3 – 3x2 – 4x + 12
g(x) = x3 + 2x2 – 25x – 50
g(x) = 2x3 + 14x2 – 2x – 14

Provide a rough sketch of g(x). Label or identify the key features on the graph.

Esmeralda is graphing a polynomial function as a parabola. Before she begins graphing it, explain how to find the vertex. Make sure you include how to determine if it will be a maximum or minimum point. Use an example quadratic function to help you explain and provide its graph.

Heinz boasts that he can predict the degree of a polynomial function just by looking at the end behavior. Can Heinz do this? Explain.

Answer 1

Both of them can be correct because a third degree polynomial can have upto three x-intercepts and a y-intercept.

For g(x) = x^3 - x^2 - 4x + 4
the zeros are 2, -2 and 1

The vertex of a parabola is the point at which the parabola turns. A parabola is maximum when the coeficient of the leading term is negative and is manimum when it is positive.