Math Problem Statement
Determine whether each of the graphs represents a quadratic function or not. If it is a quadratic function, identify its domain and range.
Solution
Let's break down the questions in the image:
8. Identifying if the table of values represents a quadratic function.
The table of values provides:
x & y \\ \hline -2 & 5 \\ -1 & 3 \\ 0 & 1 \\ 1 & 3 \\ 2 & 5 \\ 3 & 7 \\ \end{array}$$ To determine if the function is quadratic, we can calculate the first and second differences of the $$y$$-values: - First differences (differences between consecutive $$y$$-values): $$3 - 5 = -2,\quad 1 - 3 = -2,\quad 3 - 1 = 2,\quad 5 - 3 = 2,\quad 7 - 5 = 2$$ - Second differences (differences between consecutive first differences): $$-2 - (-2) = 0,\quad 2 - (-2) = 4,\quad 2 - 2 = 0$$ Since the second differences are not constant (they go from 0 to 4 to 0), this is not a quadratic function. ### D. Determine if the given graphs are quadratic functions. #### Graph 9 This graph is a parabola opening upwards. Parabolas represent quadratic functions of the form $$y = ax^2 + bx + c$$. Thus, **Graph 9 represents a quadratic function**. **Domain**: All real numbers, $$(-\infty, \infty)$$. **Range**: Since the parabola opens upwards and has a minimum at the vertex, the range is $$[0, \infty)$$. #### Graph 10 The second graph shows the equation $$y^2 = -4x$$. This is not a quadratic function because it represents a sideways parabola, which is actually a relation and not a function in terms of $$x$$. **Graph 10 does not represent a quadratic function**. --- Would you like further clarification on any of the above concepts? ### 5 Relative Questions: 1. What defines a quadratic function based on its equation? 2. How can you determine the domain and range of a quadratic function? 3. Can a sideways parabola ever represent a function? Why or why not? 4. Why are second differences used to verify if a sequence represents a quadratic function? 5. How do the properties of a quadratic graph change depending on the sign of the leading coefficient? ### Tip: Always verify if the graph of a function is quadratic by checking if it forms a parabola that opens either upwards or downwards!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Graph Analysis
Algebra
Formulas
Standard form of a quadratic function y = ax^2 + bx + c
Theorems
Quadratic Functions and Parabolas
Domain and Range Theorems
Suitable Grade Level
Grades 9-12