Understanding the Vertex of a Quadratic Equation: A Step-by-Step Guide

Have you ever looked at a quadratic equation and wondered how to find its vertex? You’re not alone! Understanding how to determine the vertex is a crucial part of mastering College Algebra, especially for those gearing up for exams. Take a closer look at the quadratic equation y = -2x² + 5x + 4. This equation takes the form of ax² + bx + c, where a, b, and c are constants.

Now, what does it mean to find the vertex? Simply put, the vertex is the point on the graph where the function either reaches its maximum or minimum value. In our example, because the coefficient of x² is negative (a = -2), we know the graph opens downwards. This means the vertex will represent the highest point on that graph. Pretty neat, huh?

So, how do we find that vertex? Here’s the formula that can make your life easier: x = -b / (2a). For our equation, b = 5 and a = -2. Plugging those values into the formula gives us:

x = -5 / (2 * -2) = -5 / -4 = 1.25.

Hold on a second! That result doesn’t seem right in terms of our choices (remember those?). Let’s focus on getting the correct vertex together.

To find the y-coordinate of the vertex, we plug that x value (1.25) back into the original equation:

y = -2(1.25)² + 5(1.25) + 4.

After performing the calculations:

• First, calculate (1.25)², which is 1.5625.
• Multiply that by -2, yielding -3.125.
• Next, calculate 5(1.25) = 6.25.
• Finally, sum these results: -3.125 + 6.25 + 4 = 7.125.

Wonderful! We’ve now found the vertex of our equation, which is located at (1.25, 7.125). But wait, we initially mentioned possible answers like (-2, 2), so let's clarify: the calculations above show how we can find vertices accurately. The choices presented seem to be a tad off based on what we just calculated.

If you are feeling a little lost, it’s completely normal! Learning how to navigate the intricacies of algebra can be like trying to find your way through a maze. It might seem tricky at first, but with practice, you’ll find that the path becomes clear.

Don’t be discouraged! Focus on the bottom line — practice makes perfect. Use practice exams to familiarize yourself with questions like these, as real exam questions can sometimes get tricky, and preparation is key.

So, as you prepare for the College Algebra CLEP, remember to keep this formula at your fingertips: x = -b / (2a). Emphasizing the process while diving into more examples and practice problems will help you get quite comfortable with these concepts. Now, how about giving those practice questions a shot? You’ve got this!