Problem 29 - 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. $$f(x)=(x-2)^{2}(x+2)$$

Step-by-Step Solution

Answer

The end behaviour of the function \(f(x)=(x-2)^{2}(x+2)\) is: as \(x \rightarrow -\infty, f(x) \rightarrow -\infty\) and as \(x \rightarrow \infty, f(x) \rightarrow \infty\). The y-intercept is -4. The x-intercepts are 2 (with multiplicity 2) and -2 (with multiplicity 1). The function has no symmetry. The function is positive on intervals (-2, 2) and (2, ∞), and negative on (-∞, -2).

Step 1: Determine End Behavior

Observe the highest degree term in the polynomial to determine the end behavior of the function. In this case, the highest degree term is \(x^3\). Since the coefficient of \(x^3\) is positive, as the value of x increases or decreases without bound, the function f(x) will tend to positive or negative infinity. That is: as \(x \rightarrow -\infty, f(x) \rightarrow -\infty\) and as \(x \rightarrow \infty, f(x) \rightarrow \infty\)

Step 2: Find the y-intercept

To find the y-intercept, substitute \(x = 0\) into the function. For the function \(f(x) = (x-2)^{2}(x+2)\), when we substitute \(x = 0\), we get \(f(0) = (-2)^{2}(-2) = -4\). Therefore, the y-intercept of the function is -4.

Step 3: Determine the x-intercepts and Multiplicities

The x-intercepts of the function are the real roots of the equation \(f(x) = 0\). For the function \(f(x) = (x-2)^{2}(x+2)\), setting \(f(x) = 0\) gives \((x-2)^{2}(x+2) = 0\). We can factor this to find \(x = 2\) and \(x = -2\). The multiplicity of 2 is 2 because the factor \((x-2)\) appears twice, while the multiplicity of -2 is 1 because the factor \((x+2)\) appears once.

Step 4: Check for Symmetry

The polynomial function can exhibit one of three types of symmetry: even, odd, or none. If it's an even function, the graph shows y-axis symmetry. If it's an odd function, the graph shows origin symmetry. If it's neither, the graph has no symmetry. In this case, the function \(f(x) = (x-2)^{2}(x+2)\) is neither even nor odd. Therefore, the graph of the function exhibits no symmetry.

Step 5: Determine Where the Function is Positive or Negative

Determine the intervals where the function is positive or negative by establishing a sign chart. Choose test values from each interval determined by the x-intercepts (i.e., (-∞, -2), (-2, 2), (2, ∞)). If when substituting the test value into the function gives a positive output, then the function is positive on that interval. If the output is negative, the function is negative on the interval. For the function, it is positive on intervals (-2, 2) and (2, ∞), and negative on (-∞, -2).

Key Concepts

End Behavior of Polynomials

Understanding the end behavior of a polynomial function is like predicting its destiny as you move towards the edges of the graph, far to the left or right.
A key tool to determine the end behavior is to look at the highest power term of the polynomial. When we have a highest power whose coefficient is positive, like in our function \(f(x)=(x-2)^2(x+2)\), with the leading term being \(x^3\), the ends of the graph will point in opposite directions (upward on the right, downward on the left) because it is an odd-degree polynomial with a positive coefficient. Mathematically, we express this as: as \(x \rightarrow -\infty, f(x) \rightarrow -\infty\) and as \(x \rightarrow \infty, f(x) \rightarrow \infty\).

It’s like a rollercoaster track that dives deep into the ground on one side and soars into the sky on the other as you extend the lines further and further away from the middle of the graph.

X-Intercepts and Multiplicities

Imagine a graph like a roadmap, and the x-intercepts are the places where the function reaches sea level. For our function \(f(x)\), the x-intercepts represent the x-values where the polynomial equals zero. These are crucial for graphing because they are where the function crosses or touches the x-axis.

The solutions to \(f(x) = 0\) give us the x-intercepts, which for our function are 2 (with a multiplicity of 2) and -2 (with a multiplicity of 1). What does this multiplicity mean? If you think about it as a bouncing ball, a larger multiplicity means that the ball bounces higher on that particular spot on the x-axis. For our function, the graph will 'bounce off' or 'touch and turn' the x-axis at \(x=2\), but it will 'cross through' the x-axis at \(x=-2\).

A rule of thumb we use here: if the multiplicity is even, the graph stays on the same side of the x-axis, while if it's odd, the graph crosses to the other side.

Intervals of Positivity and Negativity

Now that we know where our polynomial hits or skips the x-axis, it's time to check its mood between these points - is it feeling positive or negative? This means we want to find out where the function is above the x-axis (positive) and where it is below (negative).

For our function \(f(x)\), it's like a game of hot and cold. We check the intervals between the intercepts and beyond. In \(f(x)=(x-2)^2(x+2)\), we start with a negative on the far left (–∞, –2), switch to positive between the two intercepts (–2, 2), and end with a positive on the far right (2, ∞).

Why does this matter? Because this shows us the regions where the function's graph is either up above the x-axis, waving at us in positivity or diving below in negativity, similar to how a submarine submerges and resurfaces as it travels.