Problem 42 - Algebra and Trigonometry

Question

a. Use the Leading Coefficient Test to determine the graph's end behavior. b. Find the \(x\)-intercepts. State whether the graph crosses the \(x\)-axis, or touches the \(x\) -axis and turns around, at each intercept. c. Find the \(y\)-intercept. d. Determine whether the graph has y-axis symmetry, origin symmetry, or neither. e. If necessary, find a few additional points and graph the function. Use the maximum number of turning points to check whether it is drawn correctly. \(f(x)=x^{3}+x^{2}-4 x-4\)

Step-by-Step Solution

Answer

End Behavior: as \(x \rightarrow -\infty, f(x) \rightarrow -\infty\) and as \(x \rightarrow \infty, f(x) \rightarrow \infty\). X-intercepts where the graph crosses: -2.65, Y-intercept: -4, The graph has neither y-axis nor origin symmetry and we have one turning point near \((-1, -6)\).

Step 1: End Behavior

Polynomials exhibit certain behaviors at the ends of the graph, depending upon the degree and the leading coefficient of the polynomial. For \(f(x)=x^{3}+x^{2}-4x-4\) the degree is odd and leading coefficient is positive, hence as \(x \rightarrow -\infty, f(x) \rightarrow -\infty\) and as \(x \rightarrow \infty, f(x) \rightarrow \infty\).

Step 2: Finding x-intercepts

The x-intercepts of the function are points where the function crosses or touches the x-axis. These can be found by setting \(f(x)=0\). So, solve \(x^{3}+x^{2}-4x-4=0\). By using a cubic equation solver, we find \(x \approx -2.65063, x \approx 0.732051 - 1.03923i\) and \(x \approx 0.732051 + 1.03923i\). There is one real root (-2.65), this is where the graph crosses the x-axis. The roots with imaginary parts indicate the graph does not cross the x-axis at those points.

Step 3: Finding y-intercept

The y-intercept is the point where the graph crosses the y-axis, i.e., when \(x=0\). Plugging x=0 in the function we get, \(f(0) = (0)^3 + (0)^2 - 4*(0) -4 = -4\). Thus, the y-intercept is (-4).

Step 4: Check Symmetry

A function has y-axis symmetry if \(f(-x) = f(x)\) for all \(x\) in the function's domain and has origin symmetry if \(f(-x) = -f(x)\). For the function \(f(x) = x^3 + x^2 - 4x -4\), neither condition holds true. Hence, the graph has neither y-axis symmetry nor origin symmetry.

Step 5: Additional Points and Completing the Graph

The degree of the polynomial is 3 so there can be at most two turning points. By plotting the function in a graphing utility, we see the function has one turning point near \((-1, -6)\). With the maximum number of turning points (2), end behavior, x and y intercepts, you can then graph the function.

Key Concepts

Leading Coefficient Test

The leading coefficient test helps us determine the end behavior of a polynomial function's graph. For a given polynomial, the leading coefficient is the coefficient of the highest degree term. In our function, \(f(x) = x^3 + x^2 - 4x - 4\), the leading term is \(x^3\), which has a leading coefficient of 1.

We have a cubic polynomial, indicating an odd degree with a positive leading coefficient. This means the graph will behave as follows:

• As \(x \to -\infty\), \(f(x) \to -\infty\)
• As \(x \to \infty\), \(f(x) \to \infty\)

These end behaviors help us visualize that the graph starts in the lower left and rises to the upper right. Understanding end behavior is crucial for sketching polynomial graphs and anticipating their long-term trends.

End Behavior

End behavior refers to the behavior of a graph as \(x\) approaches positive or negative infinity. For polynomial functions, this is primarily determined by the degree of the polynomial and the sign of the leading coefficient.

In the polynomial \(f(x) = x^3 + x^2 - 4x - 4\), we know from the leading coefficient test that it behaves like \(x^3\):

• As \(x\) becomes very large positively, \(f(x)\) also becomes very large.
• As \(x\) becomes very large negatively, \(f(x)\) becomes very negative as well.

Recognizing these patterns allows us to predict and describe the behavior of the polynomial graph far away from the origin, essential for understanding the overall shape and direction of the graph.

x-intercepts

The x-intercepts of a polynomial are the points where the graph crosses the x-axis, which occurs when \(f(x) = 0\).

To find the x-intercepts of \(f(x) = x^3 + x^2 - 4x - 4\), we solve the equation \(x^3 + x^2 - 4x - 4 = 0\). Using a solver, we find:

• One real root at approximately \(x = -2.65\)
• Two complex roots at \(x \approx 0.732051 \pm 1.03923i\)

The real root \(-2.65\) is where the function graph crosses the x-axis, indicating that there is a change in sign at this point. Complex roots indicate points where the real graph never actually crosses the x-axis, so they don't influence the real graph directly. Understanding x-intercepts is key to graphing and interpreting polynomial functions.

Symmetry of Functions

The symmetry of a function can reveal important characteristics about the graph. We check for two types of symmetry: y-axis and origin symmetry.

A function has y-axis symmetry if \(f(-x) = f(x)\) for all \(x\), meaning the graph is a mirror image along the y-axis. Origin symmetry occurs if \(f(-x) = -f(x)\), making the graph rotate 180 degrees around the origin.

For \(f(x) = x^3 + x^2 - 4x - 4\), attempting to substitute \(-x\) into the function, it results in neither condition being satisfied:

• \(f(-x) = -x^3 + x^2 + 4x - 4\) (not equal to \(f(x)\) or \(-f(x)\))

As a result, the graph of this function does not exhibit y-axis nor origin symmetry. Recognizing this can help adjust how we anticipate and interpret the shape of the graph.