Understanding the Range of Quadratic Functions in College Algebra

When you’re navigating through the mesmerizing world of College Algebra, one question that often pops up is about the range of functions. More specifically, when considering a quadratic function like (y = 4x^2 - 3x + 8), understanding how to determine the range becomes crucial. So, what is the range of this function? Spoiler alert: it’s all real numbers! But how exactly did we arrive at that conclusion? Let’s break it down together.

First off, what does the range of a function even mean? The range encompasses all possible output values that a function can produce based on its input values. For our function at hand, which is a quadratic equation, the graph will unveil itself as a beautiful parabola. And just to clarify, a quadratic function is typically represented in the form (ax^2 + bx + c), where (a), (b), and (c) are constants. In our case, (a) is 4, (b) is -3, and (c) is 8.

Now, one might say, "Wait a minute! How can a parabola give me all real numbers for its range?" Here’s the thing—parabolas can either open upwards or downwards depending on whether the leading coefficient ((a)) is positive or negative. Since our coefficient ((4)) is positive, this means our parabola opens upward. And what happens to parabolas that open upward? They extend infinitely in the upward direction, creating all values above the vertex.

Let’s dig a bit deeper into what the vertex looks like. The vertex of a parabola is the point where it changes direction—it’s the highest or lowest point of the graph. In our case, because the parabola opens upward, the vertex serves as the lowest point. To find this point, we can use the vertex formula (x = -\frac{b}{2a}). Plugging in our (b) and (a), we get:

[ x = -\frac{-3}{2(4)} = \frac{3}{8} ]

To find the y-coordinate of the vertex, we need to substitute (x = \frac{3}{8}) back into the original function:

[ y = 4\left(\frac{3}{8}\right)^2 - 3\left(\frac{3}{8}\right) + 8 ]

When you work out the numbers, you’ll find that the y-value is slightly above 8, specifically 8.15625. So, the vertex of our parabola sits at ((\frac{3}{8}, 8.15625)).

And checking this applies to our range discussion, does this mean our range starts from 8.15625 and goes infinitely upward? Well, yes! That’s the minimum y-value of the function. The parabola then stretches upward without bounds, covering all values from that vertex and beyond.

Let’s zoom out a bit. The range here, as established, isn’t limited to just that initial vertex output but rolls out across all real numbers—like a wide-open highway stretching infinitely in both directions. So it doesn’t just stop at 8 or any other number; it extends to include every possible output.

If we circle back to the multiple-choice question we started with, it’s clear why option C—"All real numbers"—is the right choice. Let's briefly go over why the other options don’t stack up:

• Option A, 8, is merely a point on our graph but doesn’t cover everything.
• Option B, 4, is just the coefficient of the (x^2) term and bears no relation to the actual range.
• Option D, "Cannot be determined," when we’ve already calculated and verified our function’s properties, simply isn't true.

As you prepare for the College Algebra CLEP prep exam, remember that understanding the properties of functions is key. It's not just about memorizing answers; it’s about connecting the dots and grasping how these mathematical concepts work in tandem. With each equation you tackle, you're not just facing numbers but learning about how they behave—a skill that will serve you well beyond the classroom.

So next time you come across a quadratic function, you’ll feel confident saying, “Yeah, I know the range! It’s all real numbers,” just like a math whiz. Ready to tackle your algebra studies? Let’s keep those mathematical gears turning and unlock the secrets of functions one equation at a time!