Questions and Answers
Qual é a forma padronizada de uma função quadrática na forma vértice?
$ax^2 + bx + c = (a(x - h))^2 + k$
O que o discriminante ($\Delta$) de uma função quadrática determina?
O número de raízes reais da função
Qual é a condição para que uma função quadrática tenha um valor máximo ou mínimo?
$a \neq 0$
Qual é a fórmula para encontrar as coordenadas do vértice de uma parábola na forma vértice?
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Qual é o efeito de mudar o valor de $a$ em uma função quadrática na forma $y = ax^2 + bx + c$?
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Study Notes
Second Degree Functions
Second degree functions are mathematical expressions whose highest power of the variable (x) is 2. These functions are commonly referred to as quadratic functions. They can model real-world phenomena such as projectile motion, accelerating objects, and the position of oscillatory systems. In this article, we will discuss various subtopics related to second degree functions, namely vertex form, discriminant, maximum/minimum value, vertex point, and graph transformations.
Vertex Form
When analyzing quadratic functions, it is often useful to write them in vertex form. This involves completing the square by adding (\frac{b^2}{4a}) to the polynomial, then factoring the perfect square trinomial. The result is the standardized form (ax^2+bx+c=(a(x-h))^2+k), where ((h, k)) represents the vertex of the parabola. The vertex form of a quadratic function has the following properties:
- The coefficient of the squared term ((a)) determines whether the parabola opens upward or downward. If (a>0), it opens upward, and if (a<0), it opens downward.
- The term (bx) determines the y-intercept of the parabola. When (b=0), the parabola passes through the origin.
- The linear expression ((a(x-h))^2) describes the vertex of the parabola, which is located at the point ((h, k)). The term (ax^2+bx+c) represents the difference between the original equation and the shifted and scaled version of the parent function (y=(x-h)^2).
Discriminant
The discriminant of a quadratic function helps determine the number and location of the roots. For the quadratic function (f(x)=ax^2+bx+c), the discriminant ((\Delta)) is given by (\Delta=b^2-4ac). Depending on the sign of (\Delta), we can classify the roots as follows:
- If (\Delta>0), the function has two distinct roots, located at x = -b / (2a) +/- sqrt((b^2)/(4a)).
- If (\Delta=0), the function has exactly one root, located at x = -b / (2a).
- If (\Delta<0), the function does not intersect the real axis and therefore has no real roots.
These cases correspond to the three possible scenarios of the curve:
- When (\Delta>0), the parabola crosses the x-axis twice.
- When (\Delta=0), the parabola touches the x-axis once.
- When (\Delta<0), the parabola never intersects the x-axis.
Maximum/Minimum Value & Vertex Point
For a quadratic function written in vertex form, the maximum or minimum value occurs at the vertex point. The y-coordinate of the vertex corresponds to the absolute minimum or maximum depending on the sign of (a). If (a>0), then the quadratic function has a local minimum, while if (a<0), it has a local maximum. The y-coordinate of the vertex is calculated as follows:
[k=\begin{cases}a(h^2)+c &\text{(for $a>0$)} \ a(h^2)-c &\text{(for $a<0$)}\end{cases}]
Here, (h) represents the x-coordinate of the vertex.
Graph Transformations
Second degree functions can be transformed through various operations, including vertical shifts, horizontal shifts, and stretching/compression. These transformations are as follows:
- Vertical Shift: Adding or subtracting a constant term (k) to the function shifts the graph up or down by (k) units.
- Horizontal Shift: Adding or subtracting a constant term (h) to the variable (x) shifts the graph left or right by (h) units.
- Stretching/Compression: Multiplying the function by a constant factor (c) stretches or compresses the graph vertically by a factor of (c). If (c>1), it stretches upward, while if (c<1), it compresses downward. If (c=0), the graph is transformed into a single point.
In summary, second degree functions are crucial for understanding various mathematical concepts and modeling real-life scenarios. By exploring their properties, such as vertex form, discriminant, maximum/minimum value, vertex point, and graph transformations, we gain valuable insights into the behavior of these essential functions.
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