Problem 50 - College Algebra

Question

In Exercises \(49-52,\) write an equation in standard form of the parabola that has the same shape as the graph of \(f(x)-2 x^{2}\) but with the given point as the vertex. (7, 4)

Step-by-Step Solution

Answer

The equation of the parabola is \(f(x) = -2(x-7)^{2} + 4\).

Step 1: Identify the Vertex

The problem states that the vertex of the parabola is at point (7, 4). Therefore, h = 7 and k = 4.

Step 2: Identify 'a' Value

The problem states that the shape is the same as the graph of \(f(x) = -2x^{2}\), this means that the 'a' value will remain the same, hence a = -2.

Step 3: Substitute Values

Substituting the values for a, h, and k into the standard form equation \((f(x) = a(x-h)^{2} + k)\) , we get the equation of our parabola: \(f(x) = -2(x-7)^{2} + 4\).

Key Concepts

Vertex of a Parabola

The vertex of a parabola is a crucial concept in understanding its graph. It represents the highest or lowest point on the parabola, depending on the direction it opens. In the given exercise, the vertex is specified as the point (7, 4). The coordinate form of the vertex is written as (h, k), with 'h' and 'k' representing the x and y coordinates, respectively. The vertex form of a quadratic function is useful for graphing and understanding the properties of the parabola.

When you know the vertex, you can easily determine the direction in which the parabola opens and its position relative to the coordinate axis. In a standard form equation of a parabola, which is \(y = a(x-h)^{2} + k\), 'h' and 'k' will be your vertex coordinates. To put it simply, if you can identify the vertex, you’ve essentially locked in the most defining feature of the parabola’s graph.

Parabola Transformation

Once you understand the vertex, the concept of parabola transformation follows naturally. Transformations involve moving the graph around the coordinate plane without changing its shape. You achieve this through adjusting the 'h' and 'k' values in the vertex form equation of the parabola. In our exercise, moving the vertex to the point (7, 4) while maintaining the shape means we're applying a horizontal shift of 7 units to the right and a vertical shift of 4 units up from the origin.

The 'a' value in the standard form equation affects the width and direction of the parabola, but since in our exercise the value of 'a' remains -2, the width and the direction (downward in this case) do not change. Thus, the transformation occurs solely by repositioning the vertex, illustrated by substituting 'h' and 'k' into the vertex form equation.

Quadratic Functions

The term quadratic functions might sound intimidating, but they are just polynomial functions of degree two. The standard form of a quadratic function is given as \(f(x) = ax^{2} + bx + c\), where a, b, and c are constants with 'a' not equal to zero. If 'a' were zero, the function wouldn't be quadratic but linear. The parabola is the graph that represents a quadratic function, and it will always be symmetrical with a single vertex.

The value of 'a' determines if the parabola opens upwards (positive 'a') or downwards (negative 'a'). In our exercise, because 'a' equals -2, we know the parabola opens downwards. No matter how the parabola is moved around the plane, if 'a' is negative, it will always open downwards and vice versa. It’s this consistency that helps students understand and predict the behavior of quadratic functions on a graph.