Problem 15 - Algebra and Trigonometry

Question

Find the coordinates of the vertex for the parabola defined by the given quadratic function. $$ f(x)=-x^{2}-2 x+8 $$

Step-by-Step Solution

Answer

The vertex of the parabola defined by \(f(x)=-x^{2}-2x+8\) is at the coordinates (1, 5).

Step 1: Calculate the x-coordinate of the vertex

The x-coordinate, h, of the vertex of a quadratic function, \(f(x) = ax^2 + bx + c\), is given by \(h = -b/2a\). So, use this formula with \(a = -1\), and \(b = -2\) to find \(h\). Thus, \(h = - (-2) / (2*-1) = 1\).

Step 2: Calculate the y-coordinate of the vertex

The y-coordinate, k, of the vertex is determined by substituting h into the original function, that is \(k = f(h)\). Substitute \(h = 1\) into the given function to find k. So, \(k = f(1) = -(1)^{2} - 2*(1) + 8 = -1 -2 + 8 = 5\).

Step 3: Final Answer

Combine the calculated x-coordinates and y-coordinates into the vertex form, (h,k), to obtain the vertex of the parabola. So, the vertex is (1, 5).

Key Concepts

Quadratic Functions

Quadratic functions form the backbone of algebra and appear in various areas of mathematics and real-world applications. They are expressed in the general form of f(x) = ax^2 + bx + c, where a, b, and c are constants, and a ≠ 0. The graph of a quadratic function is a curve called a parabola. It has a distinctive 'U' or an upside-down 'U' shape depending on whether a is positive or negative, respectively.

The parabola opens upwards when a > 0 and downward when a < 0. Quadratic functions have many properties, including a unique vertex, which is the highest or lowest point on the graph, and an axis of symmetry that runs through the vertex dividing the parabola into two symmetrical halves. Understanding these properties helps us to sketch the parabola and solve various problems, like finding the maxima or minima of a function.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, merges algebra and geometry, allowing the study of geometric figures using a coordinate system. This method enables the precise plotting of shapes and points on the plane using ordered pairs, (x,y).

In the context of quadratic functions, coordinate geometry is used to graph parabolas, identify their vertices, axes of symmetry, and intercepts, and solve various optimization problems. The (x,y) coordinates in a parabola represent any point along the curve of the graph, and specifically, the vertex coordinates (h,k) represent the apex of the parabola. By knowing the vertex, you can easily understand the direction in which the parabola opens and its maximum or minimum value.

Vertex Form of a Parabola

The vertex form of a parabola is an insightful way to express quadratic functions. It's written as f(x) = a(x - h)^2 + k, where (h,k) is the vertex of the parabola, and a determines the direction of the opening and the width of the parabola. This form is especially convenient for graphing because it provides you directly with the vertex's coordinates.

Transitioning a quadratic function into vertex form involves completing the square or using the formula for the vertex's x-coordinate, h = -b/2a. Once you have h, you can determine the y-coordinate k by evaluating the function at h, thus finding the vertex. For example, in the given function f(x) = -x^2 - 2x + 8, once you identify h and k, you could rewrite it in the vertex form as f(x) = -(x - 1)^2 + 5, making it easier to graph and analyze the parabola.