Finding the Happy Intervals: Identifying Where Quadratic Functions are Positive

Hello there, math enthusiasts! Today, we're going to dive into the wonderful world of quadratic functions and learn how to identify the intervals on which each quadratic function is positive. Buckle up, because we're in for a ride! Guys, explore more in Guides And Explainers and identify the intervals on which each quadratic function is positive.

Understanding Quadratic Functions

First things first, let's make sure we're on the same page about what a quadratic function is. A quadratic function, or quadratic equation, is a function that can be written in the form:

f(x) = ax^2 + bx + c

where `a`, `b`, and `c` are constants, and `a` is not equal to zero (because if it were, we'd just have a linear function, not a quadratic one). The graph of a quadratic function is a parabola, which is a U-shaped curve that opens either upwards or downwards, depending on the value of `a`.

The Vertex: The Peak of the Parabola

The highest or lowest point on a parabola is called the vertex. For a quadratic function, the x-coordinate of the vertex can be found using the formula:

x = -b / (2a)

This is the x-value where the parabola reaches its peak (if `a` is positive) or valley (if `a` is negative). Now, why is the vertex important? Well, it's because the parabola is symmetric around the vertex, which means that the intervals where the function is positive (or negative) are mirror images of each other.

The Discriminant: The Key to Intervals

To find the intervals where a quadratic function is positive, we need to look at the discriminant of the quadratic equation. The discriminant is the value of `b^2 - 4ac`, and it tells us a lot about the roots of the equation (and thus, the intervals where the function is positive).

- If the discriminant is positive, the equation has two distinct real roots. This means that the parabola crosses the x-axis at two points, and the function is positive between these two roots. - If the discriminant is zero, the equation has one real root (actually, it's a double root). This means that the parabola touches the x-axis at one point, and the function is positive outside of this point. - If the discriminant is negative, the equation has no real roots. This means that the parabola does not cross the x-axis, and the function is always positive (or always negative, depending on the sign of `a`).

Interval Notation: Writing It Down

Now, let's talk about how to write down the intervals where the function is positive using interval notation. Interval notation is a way of writing down intervals using square brackets, parentheses, and the symbols `∪` (union) and `∩` (intersection).

- If the function is positive for all x, we write it as `(∞, ∞)`. - If the function is positive for all x greater than a certain number, we write it as `(a, ∞)`. - If the function is positive for all x less than a certain number, we write it as `(∞, a)`. - If the function is positive for all x between two numbers, we write it as `[a, b]` or `(a, b)` depending on whether the function is included at the endpoints or not.

Examples: Putting It All Together

Let's look at a few examples to see how to identify the intervals where a quadratic function is positive.

Example 1: A Positive Discriminant

Consider the function:

f(x) = x^2 - 4x + 3

Here, `a = 1`, `b = -4`, and `c = 3`. The discriminant is:

b^2 - 4ac = (-4)^2 - 4(1)(3) = 16 - 12 = 4

Since the discriminant is positive, we know that the function has two distinct real roots. We can find these roots by factoring the quadratic equation:

x^2 - 4x + 3 = (x - 1)(x - 3)

So the roots are `x = 1` and `x = 3`. The function is positive between these two roots, so the interval where the function is positive is:

(1, 3)

Example 2: A Zero Discriminant

Now let's look at the function:

g(x) = x^2 - 6x + 9

Here, `a = 1`, `b = -6`, and `c = 9`. The discriminant is:

b^2 - 4ac = (-6)^2 - 4(1)(9) = 36 - 36 = 0

Since the discriminant is zero, we know that the function has one real root. We can find this root by factoring the quadratic equation:

x^2 - 6x + 9 = (x - 3)^2

So the root is `x = 3`. The function is positive outside of this root, so the intervals where the function is positive are:

(-∞, 3) and (3, ∞)

Example 3: A Negative Discriminant

Finally, let's consider the function:

h(x) = x^2 + 2x - 3

Here, `a = 1`, `b = 2`, and `c = -3`. The discriminant is:

b^2 - 4ac = 2^2 - 4(1)(-3) = 4 + 12 = 16

Since the discriminant is positive, we know that the function has two distinct real roots. However, we can also see that `a = 1`, which means that the parabola opens upwards. Therefore, the function is always positive, and the interval where the function is positive is:

(-∞, ∞)

Conclusion: You're an Interval Pro Now!

And there you have it, folks! We've covered everything you need to know to identify the intervals on which each quadratic function is positive. You're now an interval pro, ready to tackle any quadratic function that comes your way.

Remember, the key is to find the roots of the quadratic equation (if any) and use them to determine the intervals where the function is positive. And don't forget, the discriminant is your friend – it's a quick and easy way to figure out what's going on with the roots.

So go forth and conquer those quadratic functions! And if you have any questions or just want to chat about math, you know where to find me. Happy calculating!