Answered

The quadratic function $f(x)=x^2$ is called the ____ of the quadratic function family. Quadratic functions can be written in different forms. The quadratic function $g(x)=x^2+3x-2$ is written in ____, $h(x)=-2(x+1)^2+3$ is written in ____, and $k(x)=(x+2)(x-4)$ is written in ____.

Verified Answer :

  • The function \(f(x) = x^2\) is the parent function.
  • The function \(g(x) = x^2 + 3x - 2\) is in standard form.
  • The function \(h(x) = -2(x + 1)^2 + 3\) is in vertex form.
  • The function \(k(x) = (x + 2)(x - 4)\) is in factored form.

Explanation

  1. Analysis of Quadratic Function Forms

Quadratic functions can be expressed in various forms, each highlighting different properties of the function's graph, such as the vertex or the x-intercepts. We will analyze each of the four given functions to identify their standard terminology.

  1. Identifying the Terminology
  2. The function \(f(x) = x^2\) is the simplest form of a quadratic function, representing the foundation from which all other parabolas are derived. In the family of quadratic functions, it is known as the parent function.
  3. The function \(g(x) = x^2 + 3x - 2\) is written in the form \(ax^2 + bx + c\), where \(a=1\), \(b=3\), and \(c=-2\). This is universally referred to as the standard form.
  4. The function \(h(x) = -2(x + 1)^2 + 3\) is written in the form \(a(x - h)^2 + k\), where the vertex of the parabola is explicitly given as \((h, k) = (-1, 3)\). This is known as the vertex form.
  5. The function \(k(x) = (x + 2)(x - 4)\) is written in the form \(a(x - r_1)(x - r_2)\), which highlights the x-intercepts of the function at \(x = -2\) and \(x = 4\). This is known as the factored form (or sometimes the intercept form).
  1. Conclusion

Based on our analysis, the blanks can be filled as follows:

  • \(f(x) = x^2\) is the parent function.
  • \(g(x) = x^2 + 3x - 2\) is in standard form.
  • \(h(x) = -2(x + 1)^2 + 3\) is in vertex form.
  • \(k(x) = (x + 2)(x - 4)\) is in factored form.

Examples

Understanding the different forms of quadratic functions is essential for real-world applications. For instance, in engineering, when designing a parabolic arch for a bridge, the vertex form might be used to easily locate the highest point of the arch. In physics, if you are modeling the trajectory of a ball, the standard form might arise from basic kinematic equations. Alternatively, the factored form is incredibly useful when trying to determine where a projectile hits the ground (the zeros). Each form acts as a different tool that simplifies specific types of problems.