According to the quadratic formula, the roots of a quadratic equation are equal to the following: Lastly, we need to examine b 2 – 4 ac, which is known as the discriminant of a quadratic equation. The answer is f( x) = –2 x 2 – 12 x – 14.īecause the parabola points downward, the value of a must be less than zero. We can use the FOIL method to evaluate ( x + 3)( x + 3). However, since the answer choices are given in standard form, not vertex form, we must expand our equation for f( x) and write it in standard form. This means that the final vertex form of the parabola is equal to f( x) = –2( x + 3) 2 + 4. We can substitute –1 in for x and –4 in for f( x). We can use the point (–1, –4), through which the parabola passes, in order to determine the value of a. In order to complete the equation for the parabola, we must find the value of a. Since the parabola has its vertex as (–3, 4), its equation in vertex form must be as follows:į( x) = a( x – (–3) 2 + 4 = a( x + 3) 2 + 4 The vertex form of a parabola is given by the following equation:į( x) = a( x – h) 2 + k, where ( h, k) is the location of the vertex, and a is a constant. In other words, only I and III (but not II) must be true.īecause we are given the vertex of the parabola, the easiest way to solve this problem will involve the use of the formula of a parabola in vertex form. We know that a and c are both less than zero, so we know choices I and III must be true however, we have just shown that b doesn't necessarily have to be less than zero. In other words, it is possible for b > 0, so it is not true that b must be less than 0. To summarize, we have just provided an example in which b is greater than zero, where f(x) has a vertex below the x-axis and a focus below the vertex. Also, because f(0) and f(1) are below the vertex, we know that the parabola opens downward, and the focus must be below the vertex.
Thus, the vertex of f(x) would be located at (1/2, –3/4), which is below the x-axis. Let's graph f(x) by trying different values of x.īecause parabolas are symmetric, the vertex must have an x-value located halfway between 0 and 1. If b = 1, and if a and c = –1, then f(x) = –x 2 + x – 1. Because we know that a and c are negative, let's assume that a and c are both –1. In other words, let's pretend that b = 1 (we are told b is not zero), and see what happens. One way to determine whether b must be negative is to assume that b is NOT negative, and see if f(x) still has a vertex below the x-axis and a focus below the vertex.
To summarize, a and c must both be less than zero. In other words, c represents the value of the y-intercept of f(x), which we have already established must be less than zero. (Any graph intersects the y-axis when x = 0.) When x = 0, f(0) = a(0) + b(0) + c = c. To find the y-coordinate of the y-intercept of f(x), we must find the value of f(x) where x = 0. Also, since the parabola points downward, it must intersect the y-axis at a point below the origin therefore, we know that the value of the y-coordinate of the y-intercept is less than zero. Since the parabola points downward, the value of a must be less than zero. The general graph of the parabola must have a shape similar to this: Because a parabola always opens toward the focus, f(x) must point downward. Going back to our original equation, l + 2(5/2) = 20, and l = 15.į(x) must be a parabola, since it contains an x 2 term. We are told that the vertex is below the x-axis, and that the focus is below the vertex. Thus, the vertex of the parabola occurs at (5/2, 25), which means that w = 5/2. We want to rewrite A(w) in the standard form of a parabola, given by f(x) = a(x-h) 2+k. The maximum value of the parabola will thus occur at the vertex. We recognize that the graph of A must be in the shape of a parabola, pointing downward. Let A be a function of w, such that A(w) = 20w - 4w 2. The length of wire used to make the pen must equal l + 2w, because this is the perimeter of a rectangle, excluding one of the lengths.
Let us assume that the side blocked by the mountain is along the length of the pen.
0 kommentar(er)
