Problem 35 - Precalculus : Building Concepts and Connections

Question

For each polynomial function, find (a) the end behavior; (b) the \(y\) -intercept; (c) the \(x\) -intercept(s) of the graph of the function and the multiplicities of the real zeros; (d) the symmetries of the graph of the function, if any; and (e) the intervals on which the function is positive or negative. Use this information to sketch a graph of the function. Factor first if the expression is not in factored form. $$f(x)=x^{3}+4 x^{2}+4 x$$

Step-by-Step Solution

Answer

End behavior as \(x\) approaches \(+\infty\) is \(+\infty\) and \(-\infty\) as \(x\) approaches \(-\infty\). The y-intercept is 0. The x-intercepts are \(x=0\) and \(x=-2\) (multiplicity 2). No symmetries are present. The function is positive for \(x < -2\) and \(x > 0\), and negative for \(-2 < x < 0\).

Step 1: End Behaviour

To find the end behavior, one needs to look at the highest degree term in the polynomial, which in this case is \(x^{3}\). Given that the exponent is odd and the leading coefficient is positive, as \(x\) approaches \(+\infty\), \(f(x)\) approaches \(+\infty\), and as \(x\) approaches \(-\infty\), \(f(x)\) approaches \(-\infty\).

Step 2: Y-intercept

The y-intercept is the value of \(f(x)\) when \(x = 0\). If we substitute \(x = 0\) into the function, it simplifies as follows: \(f(x)=0^{3}+4*0^{2}+4*0 = 0\). Therefore, the y-intercept is \(0\).

Step 3: X-intercepts and Zero Multiplicities

To find the x-intercepts, we must first set the function equal to zero. Doing so gives \(x^{3}+4 x^{2}+4 x =0\), which can be factored as \(x(x^{2}+4x+4) = 0\), further simplified into \(x(x+2)^{2}=0\). Hence, the function has zeros at x=0 (with multiplicity 1) and x=-2 (with multiplicity 2).

Step 4: Symmetry

If a function is symmetric about the y-axis, it is an even function. If it is symmetric about the origin, it is an odd function. Analysing the given function, it can be seen that it is neither even nor odd as it doesn’t satisfy the criteria for either. Hence, there are no symmetries in this function.

Step 5: Intervals of Positivity and Negativity

Given that the function has real zeros at \(x=0\) and \(x=-2\), it splits the number line into three intervals: \(-\infty < x < -2\), \(-2 < x < 0\), and \(x > 0\). Test each interval with test points, i.e., select a number in each interval and evaluate the function at that point. If the function yields a positive result, the function is positive for that interval. Similarly, if it yields a negative result, the function is negative for that interval. Here, the function is positive when \(x < -2\), negative when \(-2 < x < 0\), and positive when \(x > 0\)

Step 6: Graphing

Using the information obtained in previous steps, we can sketch the graph. The y-intercept is at 0, x-intercepts are 0 and -2, function increases as \(x\) approaches both \(-\infty\) and \(+\infty\), and finally, switch from positive to negative at roots.

Key Concepts

End Behavior of Polynomials

Understanding the end behavior of polynomials is essential for predicting the direction a function's graph will go infinitely far in the positive or negative x-direction. For a polynomial function like f(x)=x^3+4x^2+4x, the end behavior is determined by the term with the highest power, which is x^3 in this case. Because the exponent is odd and the coefficient is positive, the graph will rise to the right and fall to the left. In other words, as x approaches positive infinity, f(x) also approaches positive infinity, and as x approaches negative infinity, f(x) decreases without bound.

Finding Y-intercepts

The y-intercept of a polynomial function can be found by evaluating the function at x=0. Take f(x)=x^3+4x^2+4x; when you substitute 0 for x, the function simplifies to f(0)=0. This means that the y-intercept is the point (0,0) on the graph. A polynomial may have only one y-intercept, as it is the point where the graph crosses the vertical y-axis.

X-intercepts and Zero Multiplicities

X-intercepts occur where the function f(x) equals zero. By setting the polynomial equal to zero and factoring, we can determine these intercepts. The polynomial f(x)=x^3+4x^2+4x can be factored into x(x+2)^2=0, indicating that it has x-intercepts at x=0 and x=-2. Additionally, the intercept at x=-2 has a multiplicity of 2, which means the graph will touch the x-axis at that point but not cross it—this is indicative of the exponent of the factored term (x+2).

Graphing Polynomial Functions

Graphing polynomial functions involves plotting critical points, like the x and y intercepts, and then using information about the function's symmetry, end behavior, and intervals of positivity/negativity to complete the graph. For f(x)=x^3+4x^2+4x, there's no symmetry relative to the y-axis or origin; the y-intercept and x-intercepts are known, and we've determined the end behavior. By connecting these observations with a smooth curve, ensuring the curve behaves correctly at intercepts and in line with the end behavior, we create an accurate sketch of the function.

Intervals of Positivity and Negativity

The intervals of positivity and negativity refer to the ranges of x values where the polynomial function is above or below the x-axis. For the function f(x)=x^3+4x^2+4x, using test points reveals that the function is positive when x < -2 and when x > 0, since f(x) produces positive outputs. Between x=-2 and x=0, the function is negative. Understanding these intervals helps us predict not only where the graph will be in relation to the x-axis but also how it transitions at the x-intercepts.