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Over 1 month ago
Prompt Details
Model
gpt
Token size
81
Example input
What is the canonical form
Example output
The canonical form is a particular form of writing for a quadratic polynomial equation, ax² + bx + c, where a, b and c are real coefficients. This shape is useful because it allows you to quickly deduce important information about the parabola associated with the equation, such as the vertex and the axis of symmetry. Also, it is often used to solve optimization problems by calculating the maximum or minimum value of the quadratic function.
To find the canonical form of a quadratic polynomial equation, we need to follow these steps:
Complete the square to obtain an expression of the form a(x - h)² + k, where h and k are constants to be determined.
Write the expression in the canonical form by grouping the constant terms and the terms in x.
Take the example of the following quadratic polynomial equation: f(x) = 2x² + 8x - 6.
Complete the square:
To. Let's divide the entire equation by 2, to make the calculations easier: f(x) = 2(x² + 4x) - 6/2
b. Let's add the square of half the coefficient of x, i.e. (4/2)² = 4, inside the parenthesis:
f(x) = 2(x² + 4x + 4 - 4) - 6/2
vs. Let's rearrange the expression to get a factorized form:
f(x) = 2((x + 2)² - 4) - 3
Write the expression in canonical form:
f(x) = 2(x + 2)² - 11
Thus, the canonical form of the equation f(x) = 2x² + 8x - 6 is f(x) = 2(x + 2)² - 11.
Now that we have the canonical form, we can quickly deduce the important features of the parabola. First, the vertex of the parabola is the point (-2,-11), because the term in x is zero when x = -2. Also, the axis of symmetry of the parabola is the vertical line passing through the vertex, i.e. x = -2. Finally, the maximum value of the function is -11, because the coefficient a is positive (meaning the parabola opens upwards) and the square of any real number is positive or zero.
In conclusion, the canonical form is a useful form to represent a quadratic polynomial equation, because it allows to quickly deduce important information about the parabola associated with the equation. The method for finding the canonical form involves completing the square and writing the expression in factored form.