Graph each polynomial function. ƒ(x)=-2x⁴+7x³-4x²-4x

Verified step by step guidance

1. Identify the degree and leading coefficient of the polynomial function \(f(x) = -2x^4 + 7x^3 - 4x^2 - 4x\). The degree is 4 (since the highest power of \(x\) is 4), and the leading coefficient is \(-2\).
2. Determine the end behavior of the graph based on the degree and leading coefficient. Since the degree is even and the leading coefficient is negative, both ends of the graph will point downwards as \(x\) approaches \(\pm \infty\).
3. Find the \(x\)-intercepts by solving the equation \(f(x) = 0\), which means solving \(-2x^4 + 7x^3 - 4x^2 - 4x = 0\). Start by factoring out the greatest common factor if possible.
4. Calculate the \(y\)-intercept by evaluating \(f(0)\). This gives the point where the graph crosses the \(y\)-axis.
5. Use the information from the intercepts and end behavior to sketch the graph, and consider testing additional points or using the first and second derivatives to find critical points and inflection points for a more accurate graph.

Key Concepts

Polynomial Functions

A polynomial function is an expression consisting of variables raised to whole-number exponents and coefficients combined using addition, subtraction, and multiplication. Understanding the degree and leading coefficient helps predict the general shape and end behavior of the graph.

End Behavior of Polynomial Graphs

The end behavior describes how the graph behaves as x approaches positive or negative infinity. It depends on the degree and leading coefficient: for even-degree polynomials with a negative leading coefficient, both ends of the graph point downward.

Finding Critical Points and Shape

Critical points occur where the derivative is zero or undefined, indicating potential maxima, minima, or inflection points. Identifying these points helps in sketching the graph accurately by showing where the function changes direction.