Range Of Quadratic Function: G(x) = -x^2 + 6x - 8

Hey guys! Today, we're diving into the world of quadratic functions and tackling a common question: how to find the range of a quadratic function. Specifically, we'll be working with the function g(x) = -x² + 6x - 8. Don't worry, it's not as intimidating as it looks! By the end of this guide, you'll be able to confidently determine the range of any quadratic function and express it in interval notation. So, let's get started!

Understanding Quadratic Functions

Before we jump into the specifics of finding the range, let's quickly recap what quadratic functions are. A quadratic function is a polynomial function of degree two, meaning the highest power of the variable (in our case, 'x') is 2. The general form of a quadratic function is f(x) = ax² + bx + c, where 'a', 'b', and 'c' are constants. These constants play a crucial role in determining the shape and position of the parabola, which is the graph of a quadratic function.

The graph of a quadratic function is always a parabola. The parabola can open upwards (if 'a' is positive) or downwards (if 'a' is negative). The vertex is the turning point of the parabola – the minimum point if it opens upwards, and the maximum point if it opens downwards. The vertex is the key to finding the range of the function.

Key Takeaway: Quadratic functions are defined by their parabolic shape, determined by the coefficients a, b, and c. The vertex of the parabola is crucial for determining the range.

Identifying the Coefficients

Okay, let's get back to our specific function: g(x) = -x² + 6x - 8. The first step is to identify the coefficients 'a', 'b', and 'c'. This is pretty straightforward:

• a = -1 (the coefficient of x²)
• b = 6 (the coefficient of x)
• c = -8 (the constant term)

Notice that 'a' is negative. This tells us that the parabola opens downwards, meaning it has a maximum point (the vertex) and the range will extend downwards from that point. This is a crucial observation for understanding the range later on.

Key Takeaway: Identifying the coefficients a, b, and c is the first step. The sign of 'a' tells us whether the parabola opens upwards or downwards.

Finding the Vertex

Now, let's find the vertex of the parabola. The vertex is a point (h, k), where 'h' is the x-coordinate and 'k' is the y-coordinate. We can find the x-coordinate (h) using the following formula:

• h = -b / 2a

Let's plug in our values for 'a' and 'b':

• h = -6 / (2 * -1) = -6 / -2 = 3

So, the x-coordinate of the vertex is 3. Now, to find the y-coordinate (k), we simply substitute 'h' (which is 3) back into our original function g(x):

• k = g(3) = -(3)² + 6(3) - 8 = -9 + 18 - 8 = 1

Therefore, the vertex of our parabola is (3, 1). Since the parabola opens downwards (a < 0), this vertex represents the maximum point of the function. This means the maximum value of g(x) is 1.

Key Takeaway: The vertex (h, k) is found using the formula h = -b / 2a, and then substituting 'h' back into the original function to find 'k'.

Determining the Range

We've finally arrived at the most important part: determining the range! Remember, the range of a function is the set of all possible output values (y-values). Since our parabola opens downwards and has a maximum point at the vertex (3, 1), the function can take on any y-value less than or equal to 1.

Think of it this way: the vertex is the highest point the parabola reaches. The parabola then extends downwards infinitely. So, the y-values will go from negative infinity up to and including the y-coordinate of the vertex, which is 1.

Key Takeaway: The range is determined by the direction the parabola opens and the y-coordinate of the vertex.

Expressing the Range in Interval Notation

Now, let's express the range in interval notation. Interval notation is a way of writing sets of numbers using intervals. We use square brackets [ ] to include endpoints and parentheses ( ) to exclude endpoints. Since our range includes all values less than or equal to 1, we write it as:

• (-∞, 1]

The parenthesis next to -∞ indicates that we can never actually reach negative infinity (it's a concept, not a number). The square bracket next to 1 indicates that 1 is included in the range.

Key Takeaway: Interval notation uses parentheses and square brackets to define the range, with square brackets indicating inclusion of the endpoint.

Putting it All Together

Let's quickly recap the steps we took to find the range of g(x) = -x² + 6x - 8:

1. Identify the coefficients: a = -1, b = 6, c = -8
2. Determine the direction of the parabola: Since a < 0, the parabola opens downwards.
3. Find the vertex:
   • h = -b / 2a = 3
   • k = g(3) = 1
   • Vertex: (3, 1)
4. Determine the range: Since the parabola opens downwards and the vertex is (3, 1), the range is all y-values less than or equal to 1.
5. Express the range in interval notation: (-∞, 1]

Practice Makes Perfect

Now that you've seen how to find the range of a quadratic function, try practicing with other examples. You can change the coefficients and see how it affects the vertex and the range. Remember to always identify the direction of the parabola first – this will help you visualize the range.

Key Takeaway: Practice with different quadratic functions to solidify your understanding.

Common Mistakes to Avoid

Here are a few common mistakes people make when finding the range of quadratic functions. Avoid these pitfalls, and you'll be well on your way to mastering this concept:

• Forgetting the sign of 'a': The sign of 'a' is crucial for determining the direction the parabola opens. A positive 'a' means the parabola opens upwards, while a negative 'a' means it opens downwards.
• Incorrectly calculating the vertex: Make sure you use the correct formula for finding 'h' (-b / 2a) and substitute 'h' back into the original function to find 'k'.
• Confusing range and domain: The domain of a quadratic function is always all real numbers (-∞, ∞). The range is the set of all possible output values, which depends on the vertex and the direction of the parabola.
• Incorrect interval notation: Remember to use square brackets [ ] to include endpoints and parentheses ( ) to exclude endpoints.

Key Takeaway: Be mindful of common mistakes to ensure accurate results.

Conclusion

Finding the range of a quadratic function might seem challenging at first, but by following these steps, you can confidently tackle any quadratic function that comes your way. Remember to identify the coefficients, find the vertex, determine the direction the parabola opens, and then express the range in interval notation. With a little practice, you'll become a pro at determining the range of quadratic functions. Keep practicing, guys, and you'll ace those math problems!