Finding Intervals Where Quadratic Functions Increase And Decrease

Hey guys! Today, we're diving into the fascinating world of quadratic functions and exploring how to pinpoint those sweet spots where they're either climbing uphill (increasing) or sliding downhill (decreasing). We'll be focusing on functions in the form f(x) = ax² + bx + c, where a isn't zero (because then it wouldn't be a quadratic, would it?). So, buckle up and let's get started!

Understanding Increasing and Decreasing Functions

Before we jump into the nitty-gritty details, let's make sure we're all on the same page about what it means for a function to be increasing or decreasing. Imagine you're walking along the graph of the function from left to right. If you're walking uphill, the function is increasing. If you're walking downhill, the function is decreasing. Pretty straightforward, right?

Mathematically, we can define it like this:

• Increasing: A function f(x) is increasing on an interval if, for any two points x₁ and x₂ in the interval where x₁ < x₂, we have f(x₁) < f(x₂). In simpler terms, as x gets bigger, f(x) also gets bigger.
• Decreasing: A function f(x) is decreasing on an interval if, for any two points x₁ and x₂ in the interval where x₁ < x₂, we have f(x₁) > f(x₂). This means as x gets bigger, f(x) gets smaller.

Now that we've got the basics down, let's see how this applies to quadratic functions.

The Role of 'a' in Determining the Parabola's Shape

The first thing we need to consider is the coefficient a in our quadratic function f(x) = ax² + bx + c. This little guy plays a HUGE role in determining the shape of the parabola, which is the U-shaped curve that represents the quadratic function.

• If a > 0: The parabola opens upwards, like a smiley face. We call this a concave up parabola. Think of it as a valley – it goes down and then comes back up.
• If a < 0: The parabola opens downwards, like a frowny face. We call this a concave down parabola. Think of it as a hill – it goes up and then comes back down.

This concavity is the key to figuring out where the function is increasing and decreasing. Let's break it down for each case.

Case 1: a > 0 (Concave Up Parabola)

When a is positive, our parabola is a smiley face. It decreases until it hits its lowest point (the vertex) and then starts increasing. The vertex is the turning point of the parabola, and its x-coordinate is super important for finding our intervals of increase and decrease.

To find the x-coordinate of the vertex, we use the formula: x = -b / 2a

Let's call this x-coordinate h. So, h = -b / 2a. Now we know the parabola changes direction at x = h.

• Decreasing Interval: The function is decreasing from negative infinity up to the vertex, which is the interval (-∞, h). Imagine walking along the parabola from the left – you're going downhill until you reach the bottom of the valley.
• Increasing Interval: The function is increasing from the vertex to positive infinity, which is the interval (h, ∞). After you reach the bottom of the valley, you start walking uphill.

So, when a > 0, the function f is increasing on ( -b/2a, ∞ ) because the parabola opens upward, and to the right of the vertex, the function values increase as x increases. This is because the slope of the tangent line to the curve is positive in this interval.

Case 2: a < 0 (Concave Down Parabola)

Now, let's flip the script and consider what happens when a is negative. Our parabola is now a frowny face. It increases until it hits its highest point (the vertex) and then starts decreasing.

The vertex still plays the same crucial role, and we find its x-coordinate using the same formula: x = -b / 2a

Again, let's call this x-coordinate h. So, h = -b / 2a.

• Increasing Interval: The function is increasing from negative infinity up to the vertex, which is the interval (-∞, h). Picture walking along the parabola from the left – you're going uphill until you reach the top of the hill.
• Decreasing Interval: The function is decreasing from the vertex to positive infinity, which is the interval (h, ∞). After you reach the peak, you start walking downhill.

Putting It All Together: A Step-by-Step Guide

Okay, guys, let's recap the whole process with a handy step-by-step guide:

1. Identify a, b, and c: Look at your quadratic function f(x) = ax² + bx + c and write down the values of a, b, and c.
2. Determine the concavity: Is a positive or negative? This tells you whether the parabola opens upwards (smiley face) or downwards (frowny face).
3. Find the vertex: Calculate the x-coordinate of the vertex using the formula h = -b / 2a.
4. Determine the intervals:
   • If a > 0 (concave up):
     • Decreasing interval: (-∞, h)
     • Increasing interval: (h, ∞)
   • If a < 0 (concave down):
     • Increasing interval: (-∞, h)
     • Decreasing interval: (h, ∞)

Examples to Solidify Your Understanding

Let's walk through a couple of examples to really nail this down.

Example 1: f(x) = 2x² + 4x - 1

1. a = 2, b = 4, c = -1
2. a > 0, so the parabola opens upwards (smiley face).
3. h = -b / 2a = -4 / (2 * 2) = -1
4. • Decreasing interval: (-∞, -1)
   • Increasing interval: (-1, ∞)

Example 2: f(x) = -x² + 6x + 3

1. a = -1, b = 6, c = 3
2. a < 0, so the parabola opens downwards (frowny face).
3. h = -b / 2a = -6 / (2 * -1) = 3
4. • Increasing interval: (-∞, 3)
   • Decreasing interval: (3, ∞)

Why Does This Work? The Reasoning Behind It

You might be wondering,