Finding Increasing Interval Of F(x) = (1/2)x² + 5x + 6

Hey guys! Let's dive into a super common type of problem in algebra and calculus: figuring out where a quadratic function is increasing. We're going to break down the steps to solve this, making it crystal clear even if you're just starting out with these concepts. The function we're tackling today is f(x) = (1/2)x² + 5x + 6. Our mission is to find the interval where the graph of this function is going uphill, or in mathematical terms, where it's increasing.

Understanding Increasing Intervals

Before we jump into the nitty-gritty, let's make sure we're all on the same page about what an increasing interval actually means. Imagine you're walking along the graph of a function from left to right. If you're going uphill, the function is increasing. Mathematically, this means that as the x-values get bigger, the y-values (or f(x) values) also get bigger. For quadratic functions, which have that classic U-shape (a parabola), there's always a point where the function switches from decreasing to increasing. This turning point is super important, and it's called the vertex.

Key Concepts to Remember

• Quadratic Functions: These are functions of the form f(x) = ax² + bx + c, where a, b, and c are constants. The graph is a parabola.
• Parabola: A U-shaped curve. If a > 0, the parabola opens upwards, and if a < 0, it opens downwards.
• Vertex: The turning point of the parabola. It's the minimum point if the parabola opens upwards and the maximum point if it opens downwards.
• Increasing Interval: The interval where the function's values are increasing as x increases.
• Decreasing Interval: The interval where the function's values are decreasing as x increases.

Step-by-Step Solution

Now, let's get into the step-by-step solution for our specific function, f(x) = (1/2)x² + 5x + 6.

1. Identify the Coefficients

First things first, we need to identify the coefficients a, b, and c in our quadratic function. Comparing f(x) = (1/2)x² + 5x + 6 to the standard form f(x) = ax² + bx + c, we can see that:

• a = 1/2
• b = 5
• c = 6

2. Determine the Direction of the Parabola

Since a = 1/2, which is greater than 0, our parabola opens upwards. This means it has a minimum point (the vertex), and the function will be decreasing to the left of the vertex and increasing to the right. Knowing this is a big clue! Because a is positive, the parabola opens upwards, implying that the function decreases until it hits the vertex and increases thereafter. The fact that the parabola opens upwards is crucial, as it tells us we are looking for the interval to the right of the vertex.

3. Find the Vertex

The vertex is the key to finding our increasing interval. The x-coordinate of the vertex, often denoted as h, can be found using the formula:

h = -b / (2a)

Plugging in our values for a and b:

h = -5 / (2 * (1/2)) = -5 / 1 = -5

So, the x-coordinate of the vertex is -5. To find the y-coordinate, k, we plug this value back into our function:

k = f(-5) = (1/2)(-5)² + 5(-5) + 6 = (1/2)(25) - 25 + 6 = 12.5 - 25 + 6 = -6.5

Thus, the vertex of our parabola is at the point (-5, -6.5). This point is the minimum value of the function.

4. Determine the Increasing Interval

Since the parabola opens upwards, the function is increasing for all x-values to the right of the vertex. In other words, the function is increasing on the interval (-5, ∞). Remember, the vertex is the turning point. To the left, the function decreases, and to the right, it increases. Therefore, the increasing interval begins at the x-coordinate of the vertex and extends to infinity.

Why This Makes Sense

Think about it visually. A parabola that opens upwards decreases until it hits its lowest point (the vertex), and then it starts climbing upwards indefinitely. The x-coordinate of the vertex marks the spot where this transition happens. So, everything to the right of that x-coordinate is where the function is increasing.

The Correct Answer

Looking back at our options, the correct answer is:

• B. (-5, ∞)

This interval represents all the x-values greater than -5, which is exactly where our function is increasing.

Alternative Method: Using the Derivative

For those of you who are familiar with calculus, there's another way to find the increasing interval: using the derivative. The derivative of a function tells us about its rate of change. If the derivative is positive, the function is increasing. Let's see how this works for our function.

1. Find the Derivative

The derivative of f(x) = (1/2)x² + 5x + 6 is:

f'(x) = x + 5

2. Set the Derivative Greater Than Zero

To find where the function is increasing, we need to find where f'(x) > 0:

x + 5 > 0

3. Solve for x

Subtracting 5 from both sides, we get:

x > -5

This tells us that the function is increasing for all x-values greater than -5, which is the interval (-5, ∞). This method confirms our previous result!

Common Mistakes to Avoid

• Confusing Increasing and Decreasing: Make sure you understand the difference between an increasing and decreasing function. An increasing function goes uphill as you move from left to right, while a decreasing function goes downhill.
• Forgetting to Consider the Direction of the Parabola: If you don't know whether the parabola opens upwards or downwards, you won't be able to determine the increasing interval correctly.
• Using the Wrong Formula for the Vertex: Make sure you use the correct formula h = -b / (2a) to find the x-coordinate of the vertex.
• Not Interpreting the Interval Correctly: Remember that the increasing interval is a range of x-values, not a single point.

Practice Problems

Want to make sure you've got this down? Try these practice problems:

1. Find the increasing interval of f(x) = x² - 4x + 3.
2. Over what interval is the graph of g(x) = -2x² + 8x - 1 increasing?
3. Determine the increasing interval for h(x) = 3x² + 6x + 2.

By working through these problems, you'll solidify your understanding of how to find increasing intervals of quadratic functions. And remember, the key is to identify the vertex and the direction the parabola opens.

Conclusion

So, finding the increasing interval of a quadratic function might seem tricky at first, but once you break it down into steps, it's totally manageable! We covered identifying coefficients, determining the direction of the parabola, finding the vertex, and finally, stating the increasing interval. Plus, we even peeked at how to use calculus to confirm our answer. Remember, the vertex is your best friend in these problems. Keep practicing, and you'll become a pro at this in no time!

Hopefully, this comprehensive guide has helped you understand how to find the increasing interval of a quadratic function. Keep practicing, and you'll master this concept in no time! Remember guys, math is all about understanding the steps and applying them consistently. Good luck, and happy problem-solving!