The expression "y = x^2 - 9" describes a quadratic function with a downward shift of 9 units. The graph of this function is a parabola that opens upward with x-intercepts at "x = 3" and "x = -3," and its vertex is at the point (0, -9).
The expression "y = x^2 - 9" represents a quadratic function in terms of the relationship between the dependent variable "y" and the independent variable "x." Let's break down the expression and interpret it:
Quadratic Form: The term "x^2" indicates a quadratic relationship. This means that the relationship between "y" and "x" is characterized by a parabolic curve when plotted on a graph.
Constant Term: The term "-9" is a constant term. It shifts the entire graph vertically downward by 9 units. This constant term is often referred to as the "vertical shift" or "vertical translation."
Overall Interpretation: The expression "y = x^2 - 9" can be interpreted as a downward-shifted parabola. The shape of the parabola is determined by the "x^2" term, and the shift downward by 9 units indicates that the vertex of the parabola is moved downward on the y-axis.
Zeros or Intercepts: To find the x-intercepts (zeros) of the function, set "y = 0" and solve for "x." In this case, "x^2 - 9 = 0" can be factored as "(x - 3)(x + 3) = 0," which means that the function crosses the x-axis at "x = 3" and "x = -3."
Vertex: The vertex of the parabola is given by the values of "x" and "y" that minimize or maximize the function. In this case, the vertex is at (0, -9) because the coefficient of the "x^2" term is positive, indicating an upward-facing parabola.
Complete question:
Consider the dependent variable y. Suppose it is given that y = x² - 9. How can the given expression y = x² - 9 be interpreted, and what does it represent in terms of the relationship between the dependent variable y and the independent variable x?