Ensayos Gratis: Funcion Cuadratica
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Definition of a Quadratic Function

A function f is a quadratic function if f(x) = ax + bx + c, where a, b, and c are real numbers, with a  0.

Properties of Graphs of Quadratic Functions

1. The graph of a quadratic function f(x) = ax + bx + c is called a parabola.

2. If a > 0, the parabola opens upward; if a < 0, the parabola opens downward.

3. As  a increases, the parabola becomes narrower; as  a decreases, the parabola becomes wider.

4. The lowest point of a parabola (when a > 0) or the highest point (when a < 0) is called the vertex.

5. The domain of a quadratic function is (- ,  ), because the graph extends indefinitely to the right and to the left. If (h, k) is the vertex of the parabola, then the range of the function is [k, ) when a > 0 and (- , k] when a < 0.

6. The graph of a quadratic function is symmetric with respect to a vertical line containing the vertex. This line is called the axis of symmetry. If (h, k) is the vertex of a parabola, then the equation of the axis of symmetry is x = h.

Vertex and Intercepts

1. To determine the vertex of the graph of a quadratic function, f(x) = ax + bx + c, we can either:

a) use the method of completing the square to rewrite the function in the form

f(x) = a(x – h) + k. The vertex is (h, k). , or

b) use the formula x = to find the x-coordinate of the vertex; the

y- coordinate of the vertex can be determined by evaluating .

2. If a > 0, then the y-coordinate of the vertex represents the minimum value of the function; if a < 0, then the y-coordinate of the vertex represents the maximum value of the function.

3. To find the y-intercept of the graph of f(x) = ax + bx + c, find f(0); to find the x-intercepts, solve the quadratic equation ax + bx + c = 0.

Example. For each of the following functions, find the vertex, axis, domain, range, intercepts, and sketch the graph.

a) f(x) = -2(x – 3) + 2

b) f(x) = x - 8x + 10

c) f(x) = 2x + 12x + 17

Sol

ution. a) According to part 1 of Vertex and Intercepts, a function written in the form

f(x) = a(x – h) + k has vertex (h, k). So, f(x) = -2(x – 3) + 2 has vertex (3,2).

The axis of symmetry is a vertical line through the vertex, so its equation is x = 3.

The domain is (- ,  ). Since a < 0 (a = -2), the range is (- , 2]. (Property 5)

To find the y-intercept, we find f(0).

f(0) = -2(0 – 3) + 2 = -2(-3) + 2 = -2(9) + 2 = -18 + 2 = -16

The y-intercept is (0, -16).

To find the x-intercept, we solve the equation, -2(x – 3) + 2 = 0

-2(x – 3) + 2 = 0

-2(x - 6x + 9) + 2 = 0 Square the binomial

-2x + 12x – 18 + 2 = 0 Simplify

-2x + 12x – 16 = 0

x - 6x + 8 = 0 Divide both sides by -2

(x – 4)(x – 2) = 0 Solve by factoring

x – 4 = 0 or x – 2 = 0

x = 4 x = 2

The x-intercepts are (4,0) and (2,0).

The graph is a parabola. Since a < 0, the parabola opens downward. Since  a =  -2 = 2, the parabola is narrower than the graph of f(x) = x . (See figure 1, page 276 in ...

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