Vertex Form: Find Vertex Of Y = (1/3)(x + 7)^2 + 3

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Finding the Vertex of a Quadratic Equation Using Vertex Form

Hey guys! Today, let's dive into how to find the vertex of a quadratic equation when it's given in vertex form. Specifically, we're going to tackle the equation y = (1/3)(x + 7)² + 3. Understanding vertex form is super useful because it gives you the vertex coordinates almost instantly. So, let's break it down step by step!

Understanding Vertex Form

Before we jump into our specific equation, let's make sure we're all on the same page about what vertex form actually is. The general form of a quadratic equation in vertex form is:

y = a(x - h)² + k

Where:

  • (h, k) is the vertex of the parabola.
  • a determines the direction the parabola opens (upward if a > 0, downward if a < 0) and the width of the parabola.

So, the magic here is that the h and k values in the equation directly give us the coordinates of the vertex. Remember that the h value has a sneaky sign change because of the (x - h) part of the equation. If you see (x + h), it really means (x - (-h)), so keep that in mind!

Why is Vertex Form So Useful?

Vertex form is incredibly useful for a few key reasons:

  1. Identifying the Vertex: As we've already mentioned, the vertex coordinates are immediately apparent. This is super handy for graphing parabolas and understanding their key features.
  2. Finding the Maximum or Minimum Value: The vertex represents either the highest or lowest point on the parabola. If the parabola opens upwards (a > 0), the vertex is the minimum point. If it opens downwards (a < 0), the vertex is the maximum point. Knowing the vertex helps you quickly find these extreme values.
  3. Transformations of Parabolas: Vertex form makes it easy to see how a basic parabola (y = x²) has been transformed. The h value represents a horizontal shift, the k value represents a vertical shift, and the a value represents a vertical stretch or compression and a possible reflection.

Common Mistakes to Avoid

Before we solve our example, let's quickly touch on some common mistakes people make when working with vertex form:

  • Forgetting the Sign Change for h: This is the big one! Always remember that the h value in the vertex is the opposite of what you see inside the parentheses. If you see (x + 7)², the h value is actually -7.
  • Mixing Up h and k: Make sure you know which value is the x-coordinate (h) and which is the y-coordinate (k) of the vertex.
  • Ignoring the 'a' Value: While the a value doesn't directly tell us the vertex, it's still important. It tells us whether the parabola opens up or down and how wide or narrow it is. This affects the overall shape of the graph.

Now that we have a solid understanding of vertex form, let's get back to our equation and find its vertex!

Identifying the Vertex in Our Equation: y = (1/3)(x + 7)² + 3

Okay, let's tackle our equation: y = (1/3)(x + 7)² + 3. Our goal is to identify the vertex, which means we need to find the (h, k) values.

Let's line up our equation with the general vertex form: y = a(x - h)² + k

Now, let's break it down piece by piece:

  • a: The value of a is 1/3. This tells us the parabola opens upwards (since a is positive) and is wider than the basic parabola y = x².
  • (x + 7)²: This part is a little trickier because of the plus sign. Remember, the vertex form has (x - h)². So, we need to think of (x + 7) as (x - (-7)). This means our h value is -7.
  • + 3: This is our k value. So, k = 3.

Therefore, the vertex of the quadratic equation y = (1/3)(x + 7)² + 3 is (-7, 3).

Step-by-Step Breakdown:

Let's recap the steps we took to find the vertex:

  1. Identify the Form: Recognize that the equation is in vertex form: y = a(x - h)² + k.
  2. Find h: Look at the term inside the parentheses. Remember to take the opposite of the number being added or subtracted from x. In our case, (x + 7) means h = -7.
  3. Find k: Look at the constant term outside the parentheses. This is your k value. In our case, k = 3.
  4. Write the Vertex: Combine h and k to write the vertex as the ordered pair (h, k). So, the vertex is (-7, 3).

It's that simple! Once you get the hang of identifying h and k, finding the vertex from vertex form becomes second nature.

Graphing the Parabola (Optional)

While we've already found the vertex, let's briefly talk about how knowing the vertex helps us graph the parabola. Knowing the vertex is a huge head start in sketching the graph. Here's why:

  1. Plot the Vertex: The vertex is the central point of the parabola. Plot the point (-7, 3) on your graph.
  2. Axis of Symmetry: The parabola is symmetrical around a vertical line that passes through the vertex. This line is called the axis of symmetry, and its equation is x = h. In our case, the axis of symmetry is x = -7. Draw a dashed line at x = -7 to represent the axis of symmetry.
  3. Use the 'a' Value: The a value tells us how the parabola opens and how wide it is. Since a = 1/3 is positive, the parabola opens upwards. The smaller the absolute value of a, the wider the parabola. To plot additional points, you can use the fact that for every unit you move horizontally from the vertex, you move a units vertically. For example, if you move 1 unit to the right of the vertex (to x = -6), the corresponding y value will be 3 + (1/3)(1)² = 3 1/3. Similarly, if you move 1 unit to the left of the vertex (to x = -8), the y value will also be 3 1/3 because of the symmetry.
  4. Plot Additional Points: Plot a few more points using the symmetry of the parabola. Connect the points with a smooth curve to sketch the graph.

While we won't go through the full graphing process here, just remember that the vertex is a crucial starting point for sketching any parabola.

Conclusion

So, there you have it! Finding the vertex of a quadratic equation in vertex form is straightforward once you understand the formula y = a(x - h)² + k. Just remember to pay close attention to the sign of h, and you'll be a vertex-finding pro in no time! Understanding vertex form not only helps you identify the vertex quickly but also gives you valuable insights into the parabola's shape, direction, and transformations. Keep practicing, and you'll master this skill in no time. You got this, guys!