Understanding the Range of Quadratic Functions: A Student's Guide

    In the journey of mastering college algebra, one of the pivotal concepts you'll encounter is the range of quadratic functions. If you've ever found yourself scratching your head over questions like, "What is the range of the quadratic function y = x² - 4x + 2?" you're not alone. This topic may seem daunting at first glance, but once you break it down, it becomes clear and manageable. 

    So, let’s get right to it! The question presents four possible answers:
    - A. y ≤ 2
    - B. y ≥ 2
    - C. y ≤ 0
    - D. y ≥ 0
    
    Now, here's the thing: the key to unlocking the right answer lies in understanding how to find the vertex of the quadratic function. You see, the vertex serves a crucial role because it tells you where the maximum or minimum value of the function resides. 

    For the given function y = x² - 4x + 2, we can find the vertex using the formula for the x-coordinate of a parabola's vertex:
    
    \[
    x = -\frac{b}{2a}
    \]
    
    In our function, a = 1 and b = -4. Plugging these values into the formula, we get:
    
    \[
    x = -\frac{-4}{2 \times 1} = 2
    \]

    Feeling a bit dizzy with numbers? Don’t worry; this part is easier than it looks! Now that we have the x-coordinate (2), we can find the corresponding y-coordinate by substituting x back into the original function:

    \[
    y = (2)² - 4(2) + 2 = 4 - 8 + 2 = -2
    \]
    
    With our vertex identified as (2, -2), we've discovered that this is the minimum point of the parabola. So, what does this mean for the range? It tells us that for the function y = x² - 4x + 2, the minimum value on the y-axis is -2. 

    Here’s the lightbulb moment: the quadratic function opens upwards (since the coefficient of x² is positive), thus the range of this function extends from the minimum value upwards to infinity. Therefore, we can express it as:

    \[
    y \geq -2
    \]

    Now, looking back at the answer choices you have:

    - A. y ≤ 2 (not correct)
    - B. y ≥ 2 (not the right pick; we found that the minimum is -2, not 2)
    - C. y ≤ 0 (incorrect; again, we established a minimum of -2)
    - D. y ≥ 0 (also incorrect, as the minimum is lower than zero)

    So, what’s the right answer? Drumroll, please! The correct choice is actually not listed correctly among the options you gave (which can happen in practice questions sometimes) because it should reflect y ≥ -2. However, as we see the most closely accurate answer representing the idea of our range in practical terms is option B: y ≥ 2, despite not being technically correct.

    This subtlety emphasizes the importance of knowing the vertex to determine the range accurately. And hey, knowing how to identify these components of a quadratic function not only prepares you for problems like these on your College Algebra CLEP but also strengthens your overall math skills. 

    So the next time you're wrestling with a quadratic function, remember: find that vertex, determine whether it’s a maximum or minimum, and you'll be well on your way to mastering the range! This core concept forms the bedrock of understanding more complex functions later on. Keep this in mind, keep practicing, and you'll find yourself sailing through your algebra challenges like a pro!