Question

Without using technology, sketch the graph of each function. Label all intercepts. a) \(f(x)=x^{4}-4 x^{3}+x^{2}+6 x\) b) \(y=x^{3}+3 x^{2}-6 x-8\) c) \(y=x^{3}-4 x^{2}+x+6\) d) \(h(x)=-x^{3}+5 x^{2}-7 x+3\) e) \(g(x)=(x-1)(x+2)^{2}(x+3)^{2}\) f) \(f(x)=-x^{4}-2 x^{3}+3 x^{2}+4 x-4\)

Short answer

Sketch the graph considering intercepts and end behavior for each function.

Step-by-step solution

Identify the type of function

For each given function, determine the degree and type of polynomial. This will help in understanding the general shape of the graph.

Find the roots

Solve for the x-intercepts by setting each equation to zero. Use factoring or polynomial division where necessary.

Find the y-intercepts

To find the y-intercept of each function, substitute 0 for x in the equation and solve for y.

Determine end behavior

Analyze the leading term of each polynomial to understand how the graph behaves as x approaches \(\pm\infty\).

Plot the intercepts and sketch the graph

Use the intercepts and end behavior determined in the previous steps to sketch the graph accurately. Plot the points and connect them with a smooth curve.

Key concepts

polynomial roots

Polynomial roots are the values of x that make the polynomial equal to zero. These are also known as the x-intercepts. To find the roots of a polynomial, set the polynomial equal to zero and solve for x. Depending on the degree, you might have multiple roots.

For example, consider the polynomial function \(f(x)=x^4-4x^3+x^2+6x\). Setting \(f(x)\) to zero, \(x^4-4x^3+x^2+6x=0\). Factoring will help in finding the roots. You can factor out an x: \(x(x^3-4x^2+x+6)=0\). Now, solve for the roots of the remaining cubic polynomial by factoring further or using the Rational Root Theorem.

Each root corresponds to an x-intercept on the graph. The degree of the polynomial indicates how many roots (including multiplicity) the polynomial should have.

intercepts of polynomials

Intercepts of polynomials are crucial for graphing. There are two types: x-intercepts and y-intercepts.

• X-intercepts: These are found by setting the polynomial equal to zero and solving for x. For example, for \(y = x^3+3x^2-6x-8\), set \(x^3+3x^2-6x-8 = 0\) and solve for x.
• Y-intercept: This is found by setting x to zero and solving for y. For the same function \(y = x^3+3x^2-6x-8\), substitute \(x = 0\) to get \(y = -8\). So the y-intercept is (0, -8).

Each intercept provides a point where the graph crosses the axes, which are essential when sketching the graph.

end behavior of polynomials

The end behavior of a polynomial describes how the graph behaves as x approaches \(\pm \infty\). This behavior depends on the leading term of the polynomial.

• If the leading term's coefficient is positive and the degree is even, both ends of the graph will rise to \(\infty\).
• If the coefficient is negative and the degree is even, both ends will fall to \(-\infty\).
• If the coefficient is positive and the degree is odd, the left end will fall to \(-\infty\) and the right end will rise to \(\infty\).
• If the coefficient is negative and the degree is odd, the left end will rise to \(\infty\) and the right will fall to \(-\infty\).

For example, \(h(x) = -x^3 + 5x^2 - 7x + 3\) has a negative leading coefficient and an odd degree, so it will rise to \(\infty\) on the left and fall to \(-\infty\) on the right.

sketching polynomial graphs

Sketching polynomial graphs requires combining knowledge of roots, intercepts, and end behavior.

1. Start by finding the x- and y-intercepts.
2. Mark the intercepts on the graph.
3. Analyze the end behavior to understand how the graph will behave as x goes to \(\pm \infty\).
4. Use the roots and intercepts to guide the shape of the curve. If a root has an even multiplicity, the graph will touch the x-axis and turn around. If it has an odd multiplicity, the graph will cross the x-axis.

For instance, to sketch \(g(x) = (x-1)(x+2)^2(x+3)^2\), note the roots: 1 (odd multiplicity), -2 and -3 (even multiplicity).

The procedure ensures an accurate and smooth graph without using technology.