Math Problem Statement
Solution
Let's go through the problem step by step. The function provided is:
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 , and for this function, we have:
Since (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:
Substituting the values of and :
Now, we substitute 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.Ask a new question for Free
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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
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