Finding The Increasing Interval Of A Quadratic Function A Step-by-Step Guide

Hey everyone! Today, we're diving into a classic calculus concept: intervals of increase for functions. Specifically, we're going to tackle a quadratic function and figure out where its graph is heading upwards. This is a super important skill, not just for math class, but also for understanding how things change and grow in the real world. So, let's get started!

Problem: Identifying Increasing Intervals

Our mission, should we choose to accept it (and we do!), is to determine the interval over which the graph of the function $f(x) = \frac{1}{2}x^2 + 5x + 6$ is increasing. We're given a few options, and only one is the right fit. Let's break down how to find it. The options are:

A. ( $-6.5, \infty$ ) B. ( $-5, \infty$) C. ( $-\infty,-5$ ) D. ( $-\infty,-6.5$ )

To solve this, we'll need to remember a few key things about quadratic functions and how they behave.

Quadratic Functions and Their Graphs

First things first, let's talk about quadratic functions. These are functions of the form $f(x) = ax^2 + bx + c$, where 'a', 'b', and 'c' are constants and 'a' isn't zero. The graphs of quadratic functions are parabolas – those lovely U-shaped curves we all know and love (or at least tolerate!).

The sign of the coefficient 'a' is super important. If 'a' is positive, the parabola opens upwards, like a smiley face. If 'a' is negative, it opens downwards, like a frowny face. In our case, $a = \frac{1}{2}$, which is positive, so our parabola opens upwards. This means it has a minimum point, called the vertex.

The Vertex: Where the Magic Happens

The vertex is the turning point of the parabola. To the left of the vertex, the function is decreasing (the graph is going downwards). To the right of the vertex, the function is increasing (the graph is going upwards). This is crucial for finding our interval of increase.

So, how do we find the vertex? There's a handy formula for the x-coordinate of the vertex: $x = -\frac{b}{2a}$. Remember our function $f(x) = \frac{1}{2}x^2 + 5x + 6$? Here, $a = \frac{1}{2}$ and $b = 5$. Let's plug these values into the formula:

$x = -\frac{5}{2(\frac{1}{2})} = -\frac{5}{1} = -5$

Voila! The x-coordinate of our vertex is -5. This is the point where the parabola switches from decreasing to increasing.

Finding the Interval of Increase

Since our parabola opens upwards, it's decreasing to the left of the vertex (from negative infinity up to -5) and increasing to the right of the vertex (from -5 to positive infinity). So, the interval where the function is increasing is $(-5, \infty)$.

Looking back at our options, we see that option B, $(-5, \infty)$, is the correct answer. We did it!

Graphical Representation

To solidify our understanding, let's visualize this. Imagine the parabola opening upwards. The vertex is at x = -5. As we move from left to right along the x-axis, the graph goes down until it hits the vertex. Then, it starts climbing upwards. This upward climb represents the interval of increase.

Why Other Options Are Incorrect

It's also helpful to understand why the other options are wrong:

• A. $(-6.5, \infty)$: This interval includes the vertex, but it also includes a portion where the function is decreasing.
• C. $(-\infty, -5)$: This is the interval where the function is decreasing, not increasing.
• D. $(-\infty, -6.5)$: This interval is also part of the decreasing portion of the graph.

Key Takeaways

• Quadratic functions have parabolic graphs.
• The sign of 'a' (the coefficient of $x^2$) determines whether the parabola opens upwards (a > 0) or downwards (a < 0).
• The vertex is the turning point of the parabola.
• The x-coordinate of the vertex can be found using the formula $x = -\frac{b}{2a}$.
• For a parabola that opens upwards, the function is increasing to the right of the vertex.

Stepping it Up: Applying the Concept to Different Scenarios

Okay, now that we've conquered this problem, let's think about how we can use this knowledge in other situations. Understanding intervals of increase and decrease is super useful in a bunch of different fields.

Real-World Applications

Think about business, for example. A company might want to model its profits as a function of time. By finding the intervals of increase, they can see when their profits are growing. Or, in physics, you might analyze the height of a projectile as a function of time. The interval of increase would tell you when the projectile is going upwards.

Variations on the Theme

Now, let's consider a slightly different problem. What if we had a quadratic function where 'a' is negative? For instance, $f(x) = -x^2 + 4x - 3$. In this case, the parabola opens downwards. The vertex is still the turning point, but now the function is increasing to the left of the vertex and decreasing to the right. So, we'd need to find the x-coordinate of the vertex and then consider the interval to its left to find where the function is increasing.

Beyond Quadratic Functions

The concept of intervals of increase and decrease isn't limited to quadratic functions. We can apply it to any function! For more complex functions, we'll often use calculus (specifically, derivatives) to find these intervals. But the basic idea remains the same: we're looking for where the graph is going upwards.

Tips for Success

Here are a few tips to keep in mind when tackling these types of problems:

1. Visualize: Sketching a quick graph can be incredibly helpful. Even a rough sketch can give you a sense of the function's behavior.
2. Know the Formulas: Remember the formula for the x-coordinate of the vertex: $x = -\frac{b}{2a}$.
3. Consider the Sign of 'a': This tells you whether the parabola opens upwards or downwards.
4. Think About the Context: In real-world problems, think about what the intervals of increase and decrease mean in the given situation.

Let's Practice!

To really nail this down, let's try another example. Suppose we have the function $g(x) = \frac{1}{4}x^2 - 2x + 1$. Can you find the interval where this function is increasing? (I'll give you a hint: start by finding the vertex!).

Working through problems like this is the best way to build your understanding and confidence. Don't be afraid to make mistakes – that's how we learn! And remember, math is a journey, not a destination. So, enjoy the ride!

Final Thoughts: Mastering the Art of Intervals

So, there you have it! We've explored how to find the intervals of increase for quadratic functions, and we've seen how this concept can be applied in various scenarios. By understanding the shape of parabolas, the importance of the vertex, and the sign of the leading coefficient, you're well-equipped to tackle these problems with confidence. Remember to practice, visualize, and don't be afraid to ask questions. Keep up the great work, and you'll be a master of intervals in no time!