What Is The Vertex Form Of The Function $y = 3x^2 - 12x + 17$? Options: A. $y = 3(x-2)^2 + 5$ B. $y = 3(x-2)^2 + 17$ C. $y = 3(x-2)^2 + 11$ D. $y = 3(x+2)^2 + 5$

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Understanding the vertex form of a quadratic function is crucial for gaining insights into its key characteristics, such as the vertex, axis of symmetry, and maximum or minimum value. This article delves into the process of converting a quadratic function from its standard form to vertex form, using the example of the function $y = 3x^2 - 12x + 17$. We will explore the underlying principles and provide a step-by-step guide to help you master this essential mathematical skill.

What is the Vertex Form of a Quadratic Function?

A quadratic function, in its standard form, is expressed as $y = ax^2 + bx + c$, where a, b, and c are constants, and a ≠ 0. The vertex form, on the other hand, provides a different perspective on the same function, highlighting its vertex. The vertex form is given by:

y=a(x−h)2+ky = a(x - h)^2 + k

where:

  • (h, k) represents the vertex of the parabola.
  • a is the same leading coefficient as in the standard form, determining the parabola's direction (upward if a > 0, downward if a < 0) and its vertical stretch or compression.

The vertex form offers a direct way to identify the vertex of the parabola, which is the point where the function reaches its minimum (if a > 0) or maximum (if a < 0) value. The x-coordinate of the vertex, h, also represents the axis of symmetry, a vertical line that divides the parabola into two symmetrical halves.

Converting from Standard Form to Vertex Form: Completing the Square

The primary technique for converting a quadratic function from standard form to vertex form is called completing the square. This method involves manipulating the quadratic expression to create a perfect square trinomial, which can then be factored into a squared binomial. Let's illustrate this process with our example function, $y = 3x^2 - 12x + 17$.

Step 1: Factor out the leading coefficient from the $x^2$ and x terms.

In our case, the leading coefficient is 3. Factoring it out from the first two terms, we get:

y=3(x2−4x)+17y = 3(x^2 - 4x) + 17

This step is crucial because it ensures that the coefficient of the $x^2$ term inside the parentheses is 1, which is necessary for completing the square.

Step 2: Complete the square inside the parentheses.

To complete the square, we need to add and subtract a constant term inside the parentheses. This constant is determined by taking half of the coefficient of the x term (which is -4 in our case), squaring it, and adding and subtracting the result. Half of -4 is -2, and squaring it gives us (-2)^2 = 4. So, we add and subtract 4 inside the parentheses:

y=3(x2−4x+4−4)+17y = 3(x^2 - 4x + 4 - 4) + 17

Notice that we've added and subtracted the same value, so we haven't changed the overall expression. However, we've strategically created a perfect square trinomial within the parentheses.

Step 3: Rewrite the perfect square trinomial as a squared binomial.

The trinomial $x^2 - 4x + 4$ is a perfect square because it can be factored as $(x - 2)^2$. So, we rewrite the expression as:

y=3((x−2)2−4)+17y = 3((x - 2)^2 - 4) + 17

Now, we have a squared binomial, which is a key component of the vertex form.

Step 4: Distribute the leading coefficient and simplify.

Distribute the 3 back into the parentheses:

y=3(x−2)2−12+17y = 3(x - 2)^2 - 12 + 17

Combine the constant terms:

y=3(x−2)2+5y = 3(x - 2)^2 + 5

We have now successfully converted the function to vertex form!

Identifying the Vertex and Axis of Symmetry

From the vertex form, $y = 3(x - 2)^2 + 5$, we can directly identify the vertex as (2, 5). The x-coordinate of the vertex, 2, represents the axis of symmetry, which is the vertical line x = 2.

Since the leading coefficient a = 3 is positive, the parabola opens upwards, and the vertex (2, 5) represents the minimum point of the function. The minimum value of the function is 5, which occurs when x = 2.

Analyzing the Answer Choices

Now, let's examine the answer choices provided in the original question:

A. $y = 3(x-2)^2 + 5$ B. $y = 3(x-2)^2 + 17$ C. $y = 3(x-2)^2 + 11$ D. $y = 3(x+2)^2 + 5$

Based on our step-by-step conversion, we found that the vertex form of the function $y = 3x^2 - 12x + 17$ is $y = 3(x - 2)^2 + 5$. Therefore, the correct answer is A.

Why Other Options are Incorrect

Let's briefly discuss why the other options are incorrect:

  • Option B: $y = 3(x-2)^2 + 17$ - This option has the correct squared binomial term but an incorrect constant term. The constant term should be 5, not 17.
  • Option C: $y = 3(x-2)^2 + 11$ - Similar to option B, this option has the correct squared binomial term but an incorrect constant term. The constant term should be 5, not 11.
  • Option D: $y = 3(x+2)^2 + 5$ - This option has an incorrect squared binomial term. It should be $(x - 2)^2$, not $(x + 2)^2$. This error would result in a vertex with an x-coordinate of -2, which is incorrect.

Applications of Vertex Form

The vertex form of a quadratic function has several practical applications:

  • Optimization Problems: In optimization problems, we often need to find the maximum or minimum value of a quadratic function. The vertex form directly provides this information.
  • Graphing Quadratic Functions: The vertex form makes it easy to graph a quadratic function. We can plot the vertex and use the axis of symmetry to sketch the parabola.
  • Real-World Modeling: Quadratic functions are used to model various real-world phenomena, such as projectile motion, the shape of a suspension bridge cable, and the profit function of a business. The vertex form can help us analyze these models and extract meaningful information.

Conclusion

Converting a quadratic function from standard form to vertex form is a fundamental skill in algebra. The process of completing the square allows us to rewrite the function in a way that reveals its vertex and axis of symmetry. By mastering this technique, you can gain a deeper understanding of quadratic functions and their applications. This comprehensive guide, using the example of $y = 3x^2 - 12x + 17$, has provided a clear, step-by-step approach to converting to vertex form and identifying its key features. Remember, the vertex form not only simplifies graphing but also provides critical insights into the maximum or minimum values of the quadratic function. The ability to manipulate and interpret these forms is essential for problem-solving in various mathematical and real-world contexts. So, practice these steps diligently, and you'll be well-equipped to tackle any quadratic function challenge that comes your way. Understanding the vertex form unlocks a deeper understanding of quadratic behavior and its practical applications.