If $a(x)=2x-4$ And $b(x)=x+2$, Which Of The Following Expressions Produces A Quadratic Function?A. $ ( A B ) ( X ) (ab)(x) ( Ab ) ( X ) [/tex] B. $\left(\frac{a}{b}\right)(x)$ C. $(a-b)(x)$ D.

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Understanding the Basics of Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where a, b, and c are constants. Quadratic functions can be represented in various forms, including standard form, vertex form, and factored form.

Analyzing the Given Functions

We are given two linear functions:

a(x)=2x−4a(x) = 2x - 4

b(x)=x+2b(x) = x + 2

To determine which expression produces a quadratic function, we need to examine each option and evaluate the resulting function.

Option A: $(ab)(x)$

To find the product of two functions, we multiply them together:

(ab)(x)=a(x)b(x)=(2x−4)(x+2)(ab)(x) = a(x)b(x) = (2x - 4)(x + 2)

Expanding the product, we get:

(ab)(x)=2x2+4x−4x−8(ab)(x) = 2x^2 + 4x - 4x - 8

Simplifying the expression, we get:

(ab)(x)=2x2−8(ab)(x) = 2x^2 - 8

This is a quadratic function, as it has a degree of two.

Option B: $\left(\frac{a}{b}\right)(x)$

To find the quotient of two functions, we divide them:

(ab)(x)=a(x)b(x)=2x−4x+2\left(\frac{a}{b}\right)(x) = \frac{a(x)}{b(x)} = \frac{2x - 4}{x + 2}

This expression is not a quadratic function, as it has a degree of one.

Option C: $(a-b)(x)$

To find the difference of two functions, we subtract them:

(a−b)(x)=a(x)−b(x)=(2x−4)−(x+2)(a-b)(x) = a(x) - b(x) = (2x - 4) - (x + 2)

Simplifying the expression, we get:

(a−b)(x)=x−6(a-b)(x) = x - 6

This is a linear function, not a quadratic function.

Conclusion

Based on our analysis, the expression that produces a quadratic function is:

(ab)(x)=2x2−8(ab)(x) = 2x^2 - 8

This is the only option that results in a quadratic function.

Key Takeaways

  • A quadratic function is a polynomial function of degree two.
  • The general form of a quadratic function is $f(x) = ax^2 + bx + c$.
  • To determine if an expression is a quadratic function, we need to examine its degree.
  • The product of two linear functions can result in a quadratic function.

Final Answer

The final answer is: A\boxed{A}

Understanding Quadratic Functions

A quadratic function is a polynomial function of degree two, which means the highest power of the variable (in this case, x) is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where a, b, and c are constants. Quadratic functions can be represented in various forms, including standard form, vertex form, and factored form.

Frequently Asked Questions

Q: What is the difference between a linear and quadratic function?

A: A linear function is a polynomial function of degree one, which means the highest power of the variable (in this case, x) is one. A quadratic function, on the other hand, is a polynomial function of degree two, which means the highest power of the variable (in this case, x) is two.

Q: How do I determine if an expression is a quadratic function?

A: To determine if an expression is a quadratic function, we need to examine its degree. If the highest power of the variable (in this case, x) is two, then the expression is a quadratic function.

Q: What is the general form of a quadratic function?

A: The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where a, b, and c are constants.

Q: Can a quadratic function be represented in different forms?

A: Yes, a quadratic function can be represented in various forms, including standard form, vertex form, and factored form.

Q: What is the standard form of a quadratic function?

A: The standard form of a quadratic function is $f(x) = ax^2 + bx + c$, where a, b, and c are constants.

Q: What is the vertex form of a quadratic function?

A: The vertex form of a quadratic function is $f(x) = a(x - h)^2 + k$, where (h, k) is the vertex of the parabola.

Q: What is the factored form of a quadratic function?

A: The factored form of a quadratic function is $f(x) = (x - r_1)(x - r_2)$, where r1 and r2 are the roots of the quadratic function.

Q: How do I find the roots of a quadratic function?

A: To find the roots of a quadratic function, we need to solve the equation $f(x) = 0$. We can use various methods, including factoring, the quadratic formula, or graphing.

Q: What is the quadratic formula?

A: The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where a, b, and c are the coefficients of the quadratic function.

Q: How do I use the quadratic formula to find the roots of a quadratic function?

A: To use the quadratic formula, we need to plug in the values of a, b, and c into the formula and simplify.

Q: What is the significance of the discriminant in the quadratic formula?

A: The discriminant is the expression $b^2 - 4ac$ under the square root in the quadratic formula. If the discriminant is positive, then the quadratic function has two distinct real roots. If the discriminant is zero, then the quadratic function has one real root. If the discriminant is negative, then the quadratic function has no real roots.

Q: Can a quadratic function have complex roots?

A: Yes, a quadratic function can have complex roots. If the discriminant is negative, then the quadratic function has two complex roots.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, we can use various methods, including plotting points, using a graphing calculator, or graphing software.

Q: What is the significance of the vertex of a quadratic function?

A: The vertex of a quadratic function is the maximum or minimum point of the parabola. It is the point where the parabola changes direction.

Q: Can a quadratic function have a horizontal or vertical asymptote?

A: No, a quadratic function cannot have a horizontal or vertical asymptote. However, it can have a horizontal or vertical tangent line.

Q: How do I find the equation of a quadratic function given its graph?

A: To find the equation of a quadratic function given its graph, we need to identify the vertex and the direction of the parabola. We can then use the vertex form of a quadratic function to write the equation.

Q: Can a quadratic function be used to model real-world phenomena?

A: Yes, a quadratic function can be used to model real-world phenomena, such as the trajectory of a projectile, the motion of an object under the influence of gravity, or the growth of a population.

Q: What are some common applications of quadratic functions?

A: Some common applications of quadratic functions include:

  • Modeling the motion of an object under the influence of gravity
  • Modeling the growth of a population
  • Modeling the trajectory of a projectile
  • Modeling the stress on a beam or a bridge
  • Modeling the voltage and current in an electrical circuit

Q: How do I use quadratic functions in real-world applications?

A: To use quadratic functions in real-world applications, we need to identify the problem and the variables involved. We can then use the quadratic function to model the situation and make predictions or recommendations.

Q: What are some common mistakes to avoid when working with quadratic functions?

A: Some common mistakes to avoid when working with quadratic functions include:

  • Not checking the domain of the function
  • Not checking the range of the function
  • Not using the correct form of the function (e.g., standard form, vertex form, factored form)
  • Not using the correct method to solve the equation (e.g., factoring, quadratic formula, graphing)
  • Not checking the validity of the solutions

Q: How do I choose the correct method to solve a quadratic equation?

A: To choose the correct method to solve a quadratic equation, we need to examine the equation and determine the best method to use. We can use factoring, the quadratic formula, or graphing, depending on the situation.

Q: What are some common tools and software used to work with quadratic functions?

A: Some common tools and software used to work with quadratic functions include:

  • Graphing calculators
  • Graphing software (e.g., Desmos, GeoGebra)
  • Computer algebra systems (e.g., Mathematica, Maple)
  • Online tools and resources (e.g., Khan Academy, Wolfram Alpha)

Q: How do I use technology to work with quadratic functions?

A: To use technology to work with quadratic functions, we need to identify the tool or software to use and follow the instructions to input the function and solve the equation.

Q: What are some common challenges when working with quadratic functions?

A: Some common challenges when working with quadratic functions include:

  • Difficulty in factoring the quadratic expression
  • Difficulty in using the quadratic formula
  • Difficulty in graphing the quadratic function
  • Difficulty in interpreting the results

Q: How do I overcome common challenges when working with quadratic functions?

A: To overcome common challenges when working with quadratic functions, we need to practice and become familiar with the methods and tools used to work with quadratic functions. We can also seek help from a teacher, tutor, or online resource.

Q: What are some common misconceptions about quadratic functions?

A: Some common misconceptions about quadratic functions include:

  • Believing that a quadratic function is always a parabola
  • Believing that a quadratic function always has two real roots
  • Believing that a quadratic function always has a horizontal or vertical asymptote
  • Believing that a quadratic function always has a maximum or minimum point

Q: How do I avoid common misconceptions about quadratic functions?

A: To avoid common misconceptions about quadratic functions, we need to understand the properties and characteristics of quadratic functions and be aware of the common misconceptions.

Q: What are some common applications of quadratic functions in science and engineering?

A: Some common applications of quadratic functions in science and engineering include:

  • Modeling the motion of an object under the influence of gravity
  • Modeling the growth of a population
  • Modeling the trajectory of a projectile
  • Modeling the stress on a beam or a bridge
  • Modeling the voltage and current in an electrical circuit

Q: How do I use quadratic functions in science and engineering?

A: To use quadratic functions in science and engineering, we need to identify the problem and the variables involved. We can then use the quadratic function to model the situation and make predictions or recommendations.

Q: What are some common challenges when using quadratic functions in science and engineering?

A: Some common challenges when using quadratic functions in science and engineering include:

  • Difficulty in modeling complex phenomena
  • Difficulty in interpreting the results
  • Difficulty in making predictions or recommendations

Q: How do I overcome common challenges when using quadratic functions in science and engineering?

A: To overcome common challenges when using quadratic functions in science and engineering, we need to practice and become familiar with the methods and tools used to work with quadratic functions. We can also seek help from a teacher, tutor, or online resource.

Q: What are some common applications of quadratic functions in economics and finance?

A: Some common applications of quadratic functions in economics