Increasing Intervals & End Behavior: Polynomial Analysis

Let's dive into some polynomial analysis! We'll tackle finding where a quadratic function is increasing and then explore the end behavior of a cubic polynomial. Buckle up, math enthusiasts!

Finding the Increasing Interval of a Quadratic Function

Okay, so the first question asks: Over what interval is the function $f(x) = 2x^2 - 20x + 45$ increasing? To solve this, we need to understand how the derivative of a function relates to its increasing and decreasing intervals. Remember, guys, the derivative tells us the slope of the tangent line at any point on the curve.

Here's the plan:

1. Find the derivative of the function.
2. Determine where the derivative is positive. A positive derivative means the function is increasing.

Let's get to it!

The derivative of $f(x) = 2x^2 - 20x + 45$ is:

$f'(x) = 4x - 20$

Now, we need to find where $f'(x) > 0$. That is:

$4x - 20 > 0$

Add 20 to both sides:

$4x > 20$

Divide both sides by 4:

$x > 5$

So, the function is increasing when $x$ is greater than 5. In interval notation, this is $(5, \infty)$.

Therefore, the correct answer is D. $(5, \infty)$.

Key takeaway: The increasing interval of a function is found by determining where its derivative is positive. For a quadratic, this usually involves finding the vertex (which gives us the minimum or maximum point) and then seeing which side of the vertex the function increases.

Understanding increasing and decreasing intervals is crucial for analyzing the behavior of functions. It helps us visualize the graph and understand how the function changes as the input variable changes. In real-world applications, this can be used to model growth, decay, and optimization problems. For example, a business might want to know when its profit function is increasing to determine the best time to increase production. Or, an engineer might want to know when a function representing the stress on a bridge is decreasing to ensure its structural integrity. This concept isn't just abstract math; it has tangible implications in various fields.

To make sure we fully grasp this concept, let's consider another example. Suppose we have a function $g(x) = -x^2 + 6x - 5$. To find its increasing interval, we first find the derivative: $g'(x) = -2x + 6$. Setting this greater than zero gives us $-2x + 6 > 0$, which simplifies to $x < 3$. So, the function $g(x)$ is increasing on the interval $(-\infty, 3)$. Notice that because the coefficient of the $x^2$ term is negative, this parabola opens downward, and the function increases to the left of the vertex.

In Summary

In summary, finding the increasing interval involves taking the derivative, setting it greater than zero, and solving for x. The solution represents the interval where the function's slope is positive, indicating an upward trend. Keep an eye on the coefficient of the ${x^2}$ term in quadratics as it indicates concavity and the direction of increase/decrease.

Determining End Behavior of a Polynomial Function

Now, let's move on to the second question: What is the end behavior for the polynomial function $p(x) = x^3 - 4x^2 + 7$? End behavior refers to what happens to the function's values (i.e., $p(x)$) as $x$ approaches positive infinity $(\infty)$ and negative infinity $(-\infty)$.

The Trick:

For polynomial functions, the end behavior is determined solely by the term with the highest degree. In this case, that's the $x^3$ term.

Here's how it works:

1. Consider what happens to $x^3$ as $x$ gets very large (positive infinity).
2. Consider what happens to $x^3$ as $x$ gets very large in the negative direction (negative infinity).

Let's break it down:

• As $x \longrightarrow \infty$, $x^3 \longrightarrow \infty$. A large positive number cubed is still a large positive number.
• As $x \longrightarrow -\infty$, $x^3 \longrightarrow -\infty$. A large negative number cubed is a large negative number (because a negative times a negative times a negative is a negative).

Therefore, the end behavior of $p(x) = x^3 - 4x^2 + 7$ is:

• As $x \longrightarrow \infty$, $p(x) \longrightarrow \infty$
• As $x \longrightarrow -\infty$, $p(x) \longrightarrow -\infty$

Key takeaway: The end behavior of a polynomial is dictated by its leading term (the term with the highest power of x). The sign and exponent of the leading term determine whether the function increases or decreases as x approaches positive or negative infinity.

Understanding end behavior is extremely useful for sketching polynomial graphs. It gives you a sense of where the graph starts and ends. For instance, if you know a polynomial has a positive leading coefficient and an even degree, you know both ends of the graph point upwards. This information can help you quickly sketch a rough graph without plotting numerous points. Moreover, in applied mathematics, end behavior can model long-term trends. For example, in population models, the end behavior of a polynomial might suggest whether a population will grow indefinitely or eventually decline.

Let's consider another example: $q(x) = -2x^4 + 5x^2 - 1$. Here, the leading term is $-2x^4$. As $x \longrightarrow \infty$, $-2x^4 \longrightarrow -\infty$, because a large positive number raised to the fourth power is positive, but multiplying by -2 makes it negative. Similarly, as $x \longrightarrow -\infty$, $-2x^4 \longrightarrow -\infty$, because a large negative number raised to the fourth power is positive, and multiplying by -2 makes it negative. Thus, both ends of the graph of $q(x)$ point downwards.

In Summary

In summary, determining the end behavior involves focusing on the leading term. If the leading coefficient is positive and the degree is even, both ends go up. If the leading coefficient is positive and the degree is odd, the right end goes up, and the left end goes down. If the leading coefficient is negative and the degree is even, both ends go down. If the leading coefficient is negative and the degree is odd, the right end goes down, and the left end goes up. Mastering this simple concept provides invaluable insights into the overall behavior of polynomial functions.

Conclusion

So, there you have it! We've successfully found the increasing interval of a quadratic function by analyzing its derivative and determined the end behavior of a cubic polynomial by focusing on its leading term. Keep practicing, and you'll become a polynomial pro in no time!