Standard Form Of Quadratic Function F(x)=7(x-1)^2+3: A Step-by-Step Solution

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What is the standard form of the function f(x)=7(x-1)^2+3?

Quadratic functions are fundamental building blocks in mathematics, appearing in various applications from physics to engineering. Understanding their different forms is crucial for solving equations, graphing parabolas, and analyzing their behavior. One of the key forms is the standard form, which provides a clear representation of the function's coefficients and properties. In this article, we will delve into the concept of the standard form of a quadratic function, explore how to convert a given function into this form, and address the question of identifying the standard form of a specific function: $f(x)=7(x-1)^2+3$. We will examine the options provided and meticulously demonstrate the process of transforming the given function into its standard form, ensuring a thorough understanding of the underlying principles.

Understanding the Standard Form of a Quadratic Function

The standard form of a quadratic function is expressed as:

f(x)=ax2+bx+cf(x) = ax^2 + bx + c

where a, b, and c are constants, and a ≠ 0. This form is particularly useful because it directly reveals several key characteristics of the parabola represented by the function:

  • a: Determines the parabola's direction (opens upwards if a > 0, downwards if a < 0) and its vertical stretch or compression.
  • b: Is related to the axis of symmetry of the parabola, which is given by the equation x = -b/(2a).
  • c: Represents the y-intercept of the parabola, as f(0) = c.

The standard form allows for easy identification of these parameters, making it a valuable tool for analyzing and graphing quadratic functions. It contrasts with other forms, such as the vertex form, which highlights the vertex of the parabola, and the factored form, which reveals the function's roots or x-intercepts.

Transforming a Quadratic Function into Standard Form

Converting a quadratic function from other forms, such as the vertex form or factored form, into standard form involves algebraic manipulation. The primary technique is to expand the expression and combine like terms. Let's illustrate this with an example. Suppose we have a quadratic function in vertex form:

f(x)=a(x−h)2+kf(x) = a(x - h)^2 + k

where (h, k) is the vertex of the parabola. To convert this to standard form, we follow these steps:

  1. Expand the squared term:

    (x−h)2=x2−2hx+h2(x - h)^2 = x^2 - 2hx + h^2

  2. Multiply by a:

    a(x−h)2=ax2−2ahx+ah2a(x - h)^2 = ax^2 - 2ahx + ah^2

  3. Add k:

    f(x)=ax2−2ahx+ah2+kf(x) = ax^2 - 2ahx + ah^2 + k

  4. Rearrange to standard form:

    f(x)=ax2+(−2ah)x+(ah2+k)f(x) = ax^2 + (-2ah)x + (ah^2 + k)

Now, the function is in the standard form f(x) = ax2 + bx + c, where b = -2ah and c = ah2 + k. This process demonstrates the general method for converting from vertex form to standard form.

Similarly, if we have a quadratic function in factored form:

f(x)=a(x−r1)(x−r2)f(x) = a(x - r_1)(x - r_2)

where r1 and r2 are the roots of the function, we can convert it to standard form by expanding the product of the factors:

  1. Expand the product of the factors:

    (x−r1)(x−r2)=x2−(r1+r2)x+r1r2(x - r_1)(x - r_2) = x^2 - (r_1 + r_2)x + r_1r_2

  2. Multiply by a:

    f(x)=a[x2−(r1+r2)x+r1r2]f(x) = a[x^2 - (r_1 + r_2)x + r_1r_2]

  3. Distribute a:

    f(x)=ax2−a(r1+r2)x+ar1r2f(x) = ax^2 - a(r_1 + r_2)x + ar_1r_2

This gives us the standard form f(x) = ax2 + bx + c, where b = -a(r1 + r2) and c = ar1r2.

The ability to convert between different forms of a quadratic function is a fundamental skill in algebra, allowing us to analyze and manipulate these functions effectively. The standard form, in particular, provides a clear and concise representation that facilitates the identification of key parameters and the understanding of the parabola's behavior.

Determining the Standard Form of f(x) = 7(x - 1)² + 3

Now, let's apply the process of converting to standard form to the given function:

f(x)=7(x−1)2+3f(x) = 7(x - 1)^2 + 3

This function is currently in vertex form, where a = 7, h = 1, and k = 3. To convert it to standard form, we follow the steps outlined earlier:

  1. Expand the squared term:

    (x−1)2=x2−2(1)x+12=x2−2x+1(x - 1)^2 = x^2 - 2(1)x + 1^2 = x^2 - 2x + 1

  2. Multiply by 7:

    7(x−1)2=7(x2−2x+1)=7x2−14x+77(x - 1)^2 = 7(x^2 - 2x + 1) = 7x^2 - 14x + 7

  3. Add 3:

    f(x)=7x2−14x+7+3f(x) = 7x^2 - 14x + 7 + 3

  4. Combine like terms:

    f(x)=7x2−14x+10f(x) = 7x^2 - 14x + 10

Therefore, the standard form of the function f(x) = 7(x - 1)2 + 3 is:

f(x)=7x2−14x+10f(x) = 7x^2 - 14x + 10

This result matches option C in the given choices.

Analyzing the Answer Choices

Let's examine the provided answer choices in light of our derived standard form:

A. $f(x) = 7x^2 - 14x - 10$ This option has the correct quadratic and linear terms but the wrong constant term. It differs from our result in the sign of the constant term. The correct constant term is +10, while this option has -10. This discrepancy indicates an error in the constant term calculation.

B. $f(x) = -7x^2 - 14x - 10$ This option has incorrect quadratic term coefficient and constant term. The quadratic term coefficient should be +7, while this option has -7. Additionally, the constant term should be +10, but this option has -10. These discrepancies suggest errors in both the quadratic and constant term calculations.

C. $f(x) = 7x^2 - 14x + 10$ This option matches our derived standard form exactly. It has the correct quadratic term coefficient (+7), the correct linear term coefficient (-14), and the correct constant term (+10). This confirms that option C is the correct standard form of the given function.

D. $f(x) = -7x^2 - 14x + 10$ This option has an incorrect quadratic term coefficient. The quadratic term coefficient should be +7, while this option has -7. This discrepancy suggests an error in the quadratic term calculation.

By comparing each option to our derived standard form, we can clearly see that option C is the only one that accurately represents the function in standard form. The other options contain errors in either the quadratic term coefficient, the constant term, or both.

Conclusion

In conclusion, the standard form of the quadratic function f(x) = 7(x - 1)2 + 3 is:

f(x)=7x2−14x+10f(x) = 7x^2 - 14x + 10

This corresponds to option C in the given choices. The process of converting from vertex form to standard form involves expanding the squared term, distributing the coefficient, and combining like terms. Understanding the standard form of a quadratic function is essential for analyzing its properties, graphing its parabola, and solving related problems. This article has provided a comprehensive guide to this concept, including a step-by-step demonstration of the conversion process and a thorough analysis of the answer choices.

The standard form of a quadratic equation is a crucial concept in algebra. Mastering the conversion to standard form is a fundamental skill. This skill allows for easy identification of key parameters. Understanding these concepts helps to analyze and graph quadratic functions efficiently. The ability to transform quadratic functions into standard form. This transformation helps in solving various mathematical problems.

Quadratic functions are expressed in various forms, including standard, vertex, and factored forms. The standard form is particularly useful for identifying coefficients. These coefficients directly relate to the parabola's characteristics. Understanding the different forms of quadratic functions is essential. This understanding helps in solving quadratic equations. It also aids in graphing parabolas.

The process of converting a quadratic function to standard form involves several algebraic steps. These steps include expanding squared terms and combining like terms. This systematic approach ensures accurate conversion. Mastering this conversion is key to solving quadratic problems. It also helps in understanding function behavior. The process of transformation. This transformation is essential for various mathematical applications.

Practice Problems

To solidify your understanding, try converting the following quadratic functions to standard form:

  1. f(x)=2(x+3)2−5f(x) = 2(x + 3)^2 - 5

  2. f(x)=−3(x−2)2+1f(x) = -3(x - 2)^2 + 1

  3. f(x)=(x−4)2+6f(x) = (x - 4)^2 + 6

By working through these practice problems, you will reinforce your skills in converting quadratic functions to standard form and gain a deeper understanding of the underlying principles.