Axis Of Symmetry & Vertex: Y = 2x^2 - 12x + 21

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Finding the Axis of Symmetry and Vertex for y = 2x^2 - 12x + 21

Hey guys! Let's break down how to find the axis of symmetry and vertex of the quadratic function y = 2x^2 - 12x + 21. This is a classic problem in algebra, and once you understand the steps, it's super straightforward. We'll go through it together, step by step, so you'll be a pro in no time. Understanding these concepts is crucial for graphing parabolas and solving various optimization problems, so let's dive in!

Understanding Quadratic Functions

Before we jump into the specifics, let's recap some basics about quadratic functions. A quadratic function is a polynomial function of the second degree, generally written in the form f(x) = ax^2 + bx + c, where 'a', 'b', and 'c' are constants, and 'a' is not zero. The graph of a quadratic function is a parabola, a U-shaped curve. The parabola opens upwards if 'a' is positive and downwards if 'a' is negative. This direction is crucial because it tells us whether the vertex is a minimum or maximum point. For instance, in our function, y = 2x^2 - 12x + 21, the coefficient 'a' is 2, which is positive. This tells us our parabola opens upwards, and the vertex will be the minimum point on the graph.

The vertex is a key feature of a parabola. It is the point where the parabola changes direction – either the lowest point (minimum) if the parabola opens upwards or the highest point (maximum) if the parabola opens downwards. The vertex has coordinates (h, k), where 'h' represents the x-coordinate and 'k' represents the y-coordinate. Finding the vertex is essential for understanding the behavior and graph of the quadratic function. It essentially gives us the "turning point" of the parabola.

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. Think of it as the parabola's mirror – whatever is on one side is mirrored on the other. The equation of the axis of symmetry is always in the form x = h, where 'h' is the x-coordinate of the vertex. This line is super helpful when graphing because it gives us a central reference point. If we know one point on the parabola, we automatically know its symmetrical counterpart on the other side of the axis of symmetry. Understanding the axis of symmetry makes graphing much easier and more accurate.

Step 1: Identify a, b, and c

Okay, first things first, let's identify the coefficients a, b, and c in our function, y = 2x^2 - 12x + 21. This is super important because these values will feed into our formulas for finding the axis of symmetry and the vertex. By comparing our equation to the standard form of a quadratic equation, which is y = ax^2 + bx + c, we can easily pick out these coefficients. Make sure you pay close attention to the signs (positive or negative) – those are crucial!

In our equation, y = 2x^2 - 12x + 21:

  • The coefficient of the x^2 term, which is 'a', is 2. So, a = 2.
  • The coefficient of the x term, which is 'b', is -12. Don't forget the negative sign! b = -12.
  • The constant term, which is 'c', is 21. So, c = 21.

Now that we have correctly identified a, b, and c, we're ready to move on to the next step. Getting these values right is absolutely critical because they are the foundation for all the calculations that follow. Misidentifying any of these will throw off your final answer, so always double-check to be sure!

Step 2: Find the Axis of Symmetry

The axis of symmetry is our first goal. To find it, we use a nifty little formula: x = -b / (2a). This formula is a lifesaver and is derived from the process of completing the square on the quadratic equation. It directly gives us the x-coordinate of the vertex, which is also the equation for the axis of symmetry. You might want to jot this formula down – it's one you'll use a lot in algebra!

Now, let's plug in the values we found earlier for 'a' and 'b'. Remember, a = 2 and b = -12. So, we substitute these values into our formula:

x = -(-12) / (2 * 2)

Notice the double negative in the numerator! Let's simplify this step by step:

x = 12 / (2 * 2)

x = 12 / 4

x = 3

So, the axis of symmetry is x = 3. That wasn't too bad, right? This vertical line, x = 3, cuts our parabola perfectly in half. It's our line of symmetry, and it passes right through the vertex. Knowing the axis of symmetry is super helpful for graphing the parabola because it gives us a central reference point. We know the vertex lies on this line, and if we find any other point on the parabola, we automatically know its symmetrical counterpart.

Step 3: Find the Vertex

Next up, we're going to find the vertex of our parabola. We already know the x-coordinate of the vertex – it's the same as the axis of symmetry we just calculated! That's the beauty of the axis of symmetry; it leads us right to the vertex. In our case, we found that the axis of symmetry is x = 3, so the x-coordinate of the vertex, often denoted as 'h', is also 3. Now we just need to find the y-coordinate, often denoted as 'k'.

To find the y-coordinate of the vertex, we're going to plug the x-coordinate (h = 3) back into our original equation, y = 2x^2 - 12x + 21. This will give us the y-value that corresponds to this x-value, which is the y-coordinate of the vertex. It's like finding the height of the parabola at its turning point.

Let's substitute x = 3 into the equation:

y = 2(3)^2 - 12(3) + 21

Now, let's simplify step by step, following the order of operations (PEMDAS/BODMAS):

y = 2(9) - 36 + 21

y = 18 - 36 + 21

y = -18 + 21

y = 3

So, the y-coordinate of the vertex, 'k', is 3. Now we have both coordinates of the vertex! The vertex is at the point (3, 3). This point is super important because it tells us the minimum value of our function (since the parabola opens upwards). It's the lowest point on the graph, and everything else is built around it.

Step 4: State the Axis of Symmetry and Vertex

Alright, guys, we've done the calculations, and now it's time to state our results clearly. This step is crucial because it ties everything together and gives us the final answer in an easy-to-understand format. It’s also what your teacher or grader will be looking for, so let's make sure we present it well. We want to be crystal clear about what we've found.

So, let’s recap: we found the axis of symmetry and the vertex of the quadratic function y = 2x^2 - 12x + 21. We used the formula x = -b / (2a) to find the axis of symmetry and then plugged the x-value into the original equation to find the y-coordinate of the vertex. Now, let’s state our findings:

  • Axis of Symmetry: The axis of symmetry is the vertical line x = 3. This line divides the parabola into two symmetrical halves. Remember, it’s not just a number; it’s an equation of a line.
  • Vertex: The vertex of the parabola is the point (3, 3). This is the minimum point on the graph since the coefficient of x^2 is positive, meaning the parabola opens upwards.

By stating the axis of symmetry as x = 3 and the vertex as (3, 3), we’re communicating the key features of this quadratic function in a clear and precise manner. This is the final step, and it’s all about making sure your hard work is easily understood.

Graphing the Parabola (Optional)

If you want to take things a step further, you can graph the parabola. Knowing the axis of symmetry and vertex makes graphing much easier. Here’s how you can do it:

  1. Plot the Vertex: Start by plotting the vertex (3, 3) on the coordinate plane. This is your anchor point.
  2. Draw the Axis of Symmetry: Draw a dashed vertical line through x = 3. This line acts as a mirror for your parabola.
  3. Find Additional Points: Choose some x-values on either side of the axis of symmetry. Plug these values into the original equation to find the corresponding y-values. For example, you could choose x = 2 and x = 4.
  4. Plot the Points: Plot the points you found in the previous step. Remember to use the symmetry of the parabola. If you plot a point on one side of the axis of symmetry, you can immediately plot its symmetrical counterpart on the other side.
  5. Draw the Parabola: Sketch a smooth U-shaped curve through the points you’ve plotted. The curve should be symmetrical around the axis of symmetry and have the vertex as its minimum point.

Graphing the parabola visually confirms our calculations and gives us a complete picture of the function's behavior. It’s a great way to reinforce your understanding and see the relationship between the equation and its graphical representation.

Conclusion

So there you have it! We've successfully determined the axis of symmetry and the vertex of the function y = 2x^2 - 12x + 21. By following these steps, you can tackle any similar quadratic function. Remember to identify a, b, and c, use the formula for the axis of symmetry, and plug the x-value back into the equation to find the vertex. Keep practicing, and you'll become a master at working with parabolas! Understanding these concepts is super valuable in algebra and beyond, so great job sticking with it. You've got this!