Math Problem Statement

Consider the function f(x) = 7x^2 - 7x. a) Determine, without graphing, whether the function has a minimum value or a maximum value. b) Find the minimum or maximum value and determine where it occurs. c) Identify the function's domain and its range.

Solution

Let's go through the problem step by step. The function provided is:

f(x)=7x27xf(x) = 7x^2 - 7x

a) Determine, without graphing, whether the function has a minimum value or a maximum value.

This is a quadratic function, and its graph is a parabola. The general form of a quadratic function is f(x)=ax2+bx+cf(x) = ax^2 + bx + c, and for this function, we have:

  • a=7a = 7
  • b=7b = -7
  • c=0c = 0

Since a>0a > 0 (7 is positive), the parabola opens upwards, indicating that the function has a minimum value.

b) Find the minimum value and determine where it occurs.

To find the minimum value of a parabola, we use the formula for the vertex, which occurs at:

x=b2ax = \frac{-b}{2a}

Substituting the values of aa and bb:

x=(7)2(7)=714=12x = \frac{-(-7)}{2(7)} = \frac{7}{14} = \frac{1}{2}

Now, we substitute x=12x = \frac{1}{2} back into the function to find the minimum value:

= 7\left(\frac{1}{4}\right) - 7\left(\frac{1}{2}\right) = \frac{7}{4} - \frac{7}{2} = \frac{7}{4} - \frac{14}{4} = -\frac{7}{4}$$ So, the minimum value is $$-\frac{7}{4}$$, and it occurs at $$x = \frac{1}{2}$$. ### c) Identify the function's domain and range. - **Domain**: Since there are no restrictions (like division by zero or square roots of negative numbers), the domain is all real numbers: $$\boxed{(-\infty, \infty)}$$ - **Range**: Since the parabola opens upwards and the minimum value is $$-\frac{7}{4}$$, the range is all values greater than or equal to $$-\frac{7}{4}$$: $$\boxed{\left[-\frac{7}{4}, \infty\right)}$$ --- Do you need further explanations or details on any part of this? Let me know if you'd like me to break it down more! Here are 5 related questions you might find helpful: 1. How do you determine whether a quadratic function opens upwards or downwards? 2. What are the steps to complete the square and find the vertex of a quadratic function? 3. How can you solve a quadratic equation using the quadratic formula? 4. How do you identify the vertex and axis of symmetry of a quadratic function from its equation? 5. What is the importance of the discriminant in determining the nature of the roots of a quadratic equation? **Tip**: The vertex formula $$x = \frac{-b}{2a}$$ can be a quick and efficient way to find the axis of symmetry and the minimum or maximum value for quadratic functions.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Parabolas
Vertex Formula
Domain and Range

Formulas

Vertex formula: x = -b / (2a)

Theorems

Properties of Parabolas

Suitable Grade Level

Grades 9-11