Select The Correct Answer.The Function $h(x) = 31x^2 + 77x + 41$ Can Also Be Written As Which Of The Following?A. $y = 31x^2 + 77x - 41$ B. $y + 41 = 31x^2 + 77x$ C. $y = 31x^2 + 77x + 41$ D. $h(x) + 41 =

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Introduction

In mathematics, functions are used to represent relationships between variables. A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). The function is often represented as a mathematical expression, which can be in the form of an equation or an inequality. In this article, we will explore the representation of a given function and select the correct answer from the options provided.

Understanding the Function

The given function is h(x)=31x2+77x+41h(x) = 31x^2 + 77x + 41. This function represents a quadratic relationship between the input variable xx and the output variable h(x)h(x). The function has a leading coefficient of 31, which indicates that the parabola opens upwards. The function also has a constant term of 41, which shifts the parabola upwards.

Analyzing the Options

We are given four options to represent the function h(x)=31x2+77x+41h(x) = 31x^2 + 77x + 41. Let's analyze each option carefully.

Option A: y=31x2+77x−41y = 31x^2 + 77x - 41

This option represents a function with a leading coefficient of 31 and a constant term of -41. However, the constant term is negative, which means that the parabola opens downwards. This is not consistent with the given function, which has a positive constant term.

Option B: y+41=31x2+77xy + 41 = 31x^2 + 77x

This option represents a function with a leading coefficient of 31 and a constant term of 0. However, the constant term is not 41, which is the constant term of the given function.

Option C: y=31x2+77x+41y = 31x^2 + 77x + 41

This option represents a function with a leading coefficient of 31 and a constant term of 41. This is consistent with the given function, which has the same leading coefficient and constant term.

Option D: h(x)+41=31x2+77xh(x) + 41 = 31x^2 + 77x

This option represents a function with a leading coefficient of 31 and a constant term of 0. However, the function is represented in terms of h(x)h(x), which is not consistent with the given function.

Conclusion

Based on the analysis of the options, we can conclude that the correct answer is Option C: y=31x2+77x+41y = 31x^2 + 77x + 41. This option represents the function h(x)=31x2+77x+41h(x) = 31x^2 + 77x + 41 correctly, with the same leading coefficient and constant term.

Key Takeaways

  • A function is a relation between a set of inputs and a set of possible outputs.
  • The function can be represented as a mathematical expression, which can be in the form of an equation or an inequality.
  • The leading coefficient of a quadratic function determines the direction of the parabola.
  • The constant term of a quadratic function determines the vertical shift of the parabola.

Final Answer

Q&A: Function Representation

Introduction

In the previous article, we explored the representation of a given function and selected the correct answer from the options provided. In this article, we will continue to provide more questions and answers related to function representation.

Q1: What is a function?

A function is a relation between a set of inputs (called the domain) and a set of possible outputs (called the range). It is often represented as a mathematical expression, which can be in the form of an equation or an inequality.

Q2: What is the difference between a function and a relation?

A function is a relation where each input corresponds to exactly one output. A relation, on the other hand, can have multiple outputs for a single input.

Q3: How do you determine the direction of a parabola?

The direction of a parabola is determined by the leading coefficient of the quadratic function. If the leading coefficient is positive, the parabola opens upwards. If the leading coefficient is negative, the parabola opens downwards.

Q4: What is the significance of the constant term in a quadratic function?

The constant term in a quadratic function determines the vertical shift of the parabola. A positive constant term shifts the parabola upwards, while a negative constant term shifts the parabola downwards.

Q5: How do you represent a function in terms of another variable?

A function can be represented in terms of another variable by using a substitution. For example, if we have a function f(x)=2x2+3x+1f(x) = 2x^2 + 3x + 1, we can represent it in terms of yy by letting y=f(x)y = f(x).

Q6: What is the difference between a function and an equation?

A function is a relation between a set of inputs and a set of possible outputs, while an equation is a statement that two expressions are equal. A function can be represented as an equation, but not all equations are functions.

Q7: How do you determine if a relation is a function?

A relation is a function if each input corresponds to exactly one output. We can determine this by checking if the relation satisfies the vertical line test.

Q8: What is the vertical line test?

The vertical line test is a method used to determine if a relation is a function. We draw a vertical line on the graph of the relation and check if it intersects the graph at more than one point. If it does, the relation is not a function.

Q9: How do you represent a function in a different coordinate system?

A function can be represented in a different coordinate system by using a transformation. For example, if we have a function f(x)=2x2+3x+1f(x) = 2x^2 + 3x + 1 and we want to represent it in terms of yy and xx, we can use the transformation y=f(x)y = f(x).

Q10: What is the significance of the domain and range of a function?

The domain and range of a function are the sets of all possible input and output values, respectively. They are important because they determine the behavior of the function and its graph.

Conclusion

In this article, we provided answers to frequently asked questions related to function representation. We hope that this article has been helpful in clarifying the concepts of function representation and will serve as a useful resource for students and educators alike.

Key Takeaways

  • A function is a relation between a set of inputs and a set of possible outputs.
  • The leading coefficient of a quadratic function determines the direction of the parabola.
  • The constant term of a quadratic function determines the vertical shift of the parabola.
  • A function can be represented in terms of another variable by using a substitution.
  • The vertical line test is a method used to determine if a relation is a function.

Final Answer

The final answer is that a function is a relation between a set of inputs and a set of possible outputs, and it can be represented in various ways, including in terms of another variable and in different coordinate systems.