An explicit equation for a quadratic function is created by expressing it in the standard form: y = ax² + bx + c, where 'a', 'b', and 'c' are constants, and 'x' is the independent variable. This form clearly shows how the dependent variable 'y' is calculated based on the value of 'x'.
Understanding the Standard Form
The standard form of a quadratic equation is essential for making it explicit because it directly reveals the relationship between the variables. Here's a breakdown:
- y: The dependent variable, representing the output of the function.
- x: The independent variable, the input of the function.
- a: The coefficient of the x² term, which determines the direction and width of the parabola.
- If 'a' is positive, the parabola opens upwards.
- If 'a' is negative, the parabola opens downwards.
- The absolute value of 'a' controls how "wide" or "narrow" the parabola is.
- b: The coefficient of the x term, which affects the parabola's position on the coordinate plane.
- c: The constant term, which represents the y-intercept of the parabola (the point where the parabola crosses the y-axis).
How to Create an Explicit Equation
To make an explicit equation, you need to:
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Determine the values of 'a', 'b', and 'c'. This can be done through different methods:
- From a Graph: You can sometimes extract the values by identifying specific points on the parabola, such as the y-intercept (which gives you 'c') and other known points, and then solving the resulting equations.
- From Given Conditions: Sometimes, you'll be provided with conditions like "the vertex is at (h,k)" or "the parabola passes through points (x₁, y₁) and (x₂, y₂)", which can be used to solve for a, b, and c.
- From a Word Problem: The problem might give you a real-world scenario that will help you figure out the relationship between x and y and help you determine the values of a,b and c.
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Substitute the values into the standard form: Once you have determined the constants, substitute the values for a, b, and c into the form y = ax² + bx + c.
Example
Let's create an explicit equation for a quadratic function:
Suppose we know that the values of a=3, b=-4 and c=10. We can create an explicit quadratic function by substituting the values of a,b and c into the standard equation.
- Standard Form: y = ax² + bx + c
- Substituting the values: y = 3x² + (-4)x + 10
- Simplifying: y = 3x² - 4x + 10
Therefore, an explicit equation for a quadratic function with a=3, b=-4 and c=10 is y = 3x² - 4x + 10.
Key Characteristics of Explicit Quadratic Equations
- Explicitly Defined Relationship: The equation directly shows how 'y' is calculated from 'x'.
- Clear Coefficients: It presents the coefficients 'a', 'b', and 'c' directly in the formula, which allows for easy manipulation and analysis.
- Easy to Use: It's simple to plug in values for 'x' to find corresponding values for 'y'.
- Graphing: It enables us to plot quadratic functions by easily generating pairs of (x,y) values which can be used to plot a parabola.
By understanding the standard form y = ax² + bx + c and following the steps to determine 'a', 'b', and 'c', you can easily make an explicit equation for any quadratic function.
