Explanation
Quadratic functions are second-degree functions, usually represented as f(x)=ax^2 + bx + c. Here, a, b and c are real numbers, with a \neq 0 required. These functions can be converted into various forms such as standard form, vertex form and factored form. To solve quadratic equations, the quadratic formula is often used and with the method of one also obtains the vertex form.
Procedure
Function term: The expression of a quadratic function in standard form.
Example: f(x) = 2x^2 + 3x - 5 shows a function term. Here, a=2,\, b=3,\, c=-5.
Standard form: The standard representation of the quadratic function.
Example: f(x) = x^2 + 4x + 4 is written in standard form. Since all terms are already present, no transformation is needed here.
Vertex form: This form directly shows the vertex of the parabola. Conversion is done via completing the square.
Example: Convert f(x) = 2x^2 + 8x + 6 to vertex form.
First, combine the quadratic term: f(x) = 2(x^2 + 4x) + 6. Complete the square: x^2 + 4x = (x+2)^2 - 4. Thus f(x) = 2(x+2)^2 - 2.
Factored form: With this form, the intersection points of the parabola with the x-axis can be read off.
Example: f(x) = x^2 - 5x + 6 has zeros at 2 and 3. Thus the function can bef(x) = (x-2)(x-3) written.
Quadratic formula: The quadratic formula is used to solve quadratic equations.
Example: Solve the equation 0 = 2x^2 + 8x + 6. First calculate the discriminant: \Delta = 16.
Substitute into the quadratic formula: x = \frac{-8 \pm 4}{4}, also x = -1 und x = -3.
Misunderstandings
- ★It is often forgotten that a \neq 0 must hold.
- ★An error is often made in completing the square by determining the term to be added incorrectly.
- ★The expression under the square root in the quadratic formula is often calculated incorrectly.
Problems
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