Select The Correct Answer.Four Equivalent Forms Of A Quadratic Function Are Given. Which Form Displays The \[$y\$\]-intercept Of The Function?A. \[$f(x)=3\left(x^2+2x-48\right)\$\]B. \[$f(x)=3x^2+6x-144\$\]C.

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Understanding Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in various forms. These forms are equivalent, meaning they represent the same function, but they can be more or less convenient to work with depending on the situation. In this article, we will explore four equivalent forms of a quadratic function and determine which form displays the y-intercept of the function.

The Four Equivalent Forms of a Quadratic Function

A. Factored Form: f(x) = 3(x^2 + 2x - 48)

The factored form of a quadratic function is written as f(x) = a(x - r)(x - s), where r and s are the roots of the function. In this case, the factored form is f(x) = 3(x^2 + 2x - 48). However, this form does not display the y-intercept of the function.

B. Standard Form: f(x) = 3x^2 + 6x - 144

The standard form of a quadratic function is written as f(x) = ax^2 + bx + c, where a, b, and c are constants. In this case, the standard form is f(x) = 3x^2 + 6x - 144. This form does not display the y-intercept of the function either.

C. Vertex Form: f(x) = 3(x - 6)^2 - 72

The vertex form of a quadratic function is written as f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the function. In this case, the vertex form is f(x) = 3(x - 6)^2 - 72. This form also does not display the y-intercept of the function.

D. General Form: f(x) = 3(x^2 + 2x - 48)

The general form of a quadratic function is written as f(x) = ax^2 + bx + c, where a, b, and c are constants. In this case, the general form is f(x) = 3(x^2 + 2x - 48). This form displays the y-intercept of the function.

Why the General Form Displays the Y-Intercept

The general form of a quadratic function, f(x) = ax^2 + bx + c, displays the y-intercept of the function because the y-intercept is the point where the function crosses the y-axis. In other words, it is the value of the function when x = 0. In the general form, the y-intercept is the constant term, c. In this case, the y-intercept is -144.

Conclusion

In conclusion, the correct form of a quadratic function that displays the y-intercept of the function is the general form, f(x) = 3(x^2 + 2x - 48). This form is the most convenient to work with when the y-intercept is needed.

Key Takeaways

  • The factored form, standard form, and vertex form of a quadratic function do not display the y-intercept of the function.
  • The general form of a quadratic function, f(x) = ax^2 + bx + c, displays the y-intercept of the function.
  • The y-intercept is the value of the function when x = 0.

Real-World Applications

Understanding the different forms of a quadratic function and how to identify the y-intercept is crucial in various real-world applications, such as:

  • Physics: When modeling the motion of an object, the y-intercept represents the initial position of the object.
  • Engineering: When designing a system, the y-intercept represents the initial condition of the system.
  • Economics: When modeling economic systems, the y-intercept represents the initial condition of the system.

Final Thoughts

Frequently Asked Questions About Quadratic Functions

Quadratic functions are a type of polynomial function that can be written in various forms. These forms are equivalent, meaning they represent the same function, but they can be more or less convenient to work with depending on the situation. In this article, we will answer some frequently asked questions about quadratic functions.

Q: What is a quadratic function?

A: A quadratic function is a type of polynomial function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants.

Q: What are the different forms of a quadratic function?

A: There are four main forms of a quadratic function:

  • Factored form: f(x) = a(x - r)(x - s), where r and s are the roots of the function.
  • Standard form: f(x) = ax^2 + bx + c, where a, b, and c are constants.
  • Vertex form: f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the function.
  • General form: f(x) = ax^2 + bx + c, where a, b, and c are constants.

Q: Which form displays the y-intercept of the function?

A: The general form of a quadratic function, f(x) = ax^2 + bx + c, displays the y-intercept of the function.

Q: What is the y-intercept of a quadratic function?

A: The y-intercept of a quadratic function is the value of the function when x = 0. It is the point where the function crosses the y-axis.

Q: How do I find the y-intercept of a quadratic function?

A: To find the y-intercept of a quadratic function, you can substitute x = 0 into the function and solve for y.

Q: What are the real-world applications of quadratic functions?

A: Quadratic functions have many real-world applications, including:

  • Physics: Modeling the motion of an object.
  • Engineering: Designing systems.
  • Economics: Modeling economic systems.

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use the following steps:

  1. Find the vertex: Find the vertex of the function by using the formula x = -b/2a.
  2. Find the x-intercepts: Find the x-intercepts of the function by setting y = 0 and solving for x.
  3. Plot the points: Plot the points (x, y) on a coordinate plane.
  4. Draw the graph: Draw the graph of the function by connecting the points.

Q: What are the different types of quadratic functions?

A: There are several types of quadratic functions, including:

  • Monic quadratic function: A quadratic function of the form f(x) = x^2 + bx + c.
  • Nonmonic quadratic function: A quadratic function of the form f(x) = ax^2 + bx + c, where a ≠ 1.
  • Quadratic function with a negative leading coefficient: A quadratic function of the form f(x) = -ax^2 + bx + c.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

  1. Write the equation: Write the quadratic equation in the form ax^2 + bx + c = 0.
  2. Factor the equation: Factor the equation, if possible.
  3. Use the quadratic formula: Use the quadratic formula x = (-b ± √(b^2 - 4ac)) / 2a to solve for x.
  4. Check the solutions: Check the solutions to make sure they are valid.

Conclusion

In conclusion, quadratic functions are a type of polynomial function that can be written in various forms. Understanding the different forms of a quadratic function and how to identify the y-intercept is crucial in various real-world applications.