Question

THOUGHT PROVOKING A jump on a pogo stick with a conventional spring can be modeled by f(x) = ?0.5(x ? 6)2 + 18, where x is the horizontal distance (in inches) and f(x) is the vertical distance (in inches). Write at least one transformation of the function and provide a possible reason for your transformation.

Short answer

A possible transformation of the function is \(g(x) = -0.5(x-8)^2 + 18\), which represents a 2-inch shift to the right from the original jump position.

Step-by-step solution

Understand the Base Function

The base function given is \(f(x) = -0.5(x - 6)^2 + 18\). This is a parabolic function (a quadratic function), which opens downwards due to the negative coefficient of \(x^2\). The vertex of this parabola, i.e., the highest point of the jump, is at (6,18). That is 6 inches horizontally and 18 inches vertically.

Define a Transformation

One possible transformation could be a horizontal shift. A shift to the right by 2 units can be represented mathematically as: \(g(x) = -0.5(x-6-2)^2 + 18 = -0.5(x-8)^2 + 18.\) This new function has its vertex at (8,18)

Interpret the Transformation

The transformation can be interpreted as a horizontal shift in the pogo stick jump. This could happen if the jumper starts the jump 2 inches further to the right than the original position. Therefore, instead of the highest point of the jump being 6 inches to the right of the starting position, it will be 8 inches to the right.

Key concepts

Graph Transformations

Graph transformations are changes made to a graph's equation that shift, stretch, shrink, or reflect the graph on the coordinate plane. In the context of quadratic functions, like the given function for a pogo stick jump, transformations can help us visualize how the graph changes under different conditions.

For instance, if we apply a horizontal shift to a quadratic function, we move the entire parabola left or right. In our example, shifting the graph to the right by 2 units transforms the base function \[f(x) = -0.5(x - 6)^2 + 18\] to \[g(x) = -0.5(x - 8)^2 + 18.\] This means the vertex, or the peak of the curve, also shifts from (6, 18) to (8, 18).

Such transformations allow us to model real-world scenarios more accurately. A horizontal shift might represent a change in starting position, like a jumper beginning a leap from a different spot.

Parabolas

Parabolas are U-shaped curves commonly seen in quadratic functions. They have unique properties that define their shape and orientation. In a quadratic function like \(f(x) = -0.5(x - 6)^2 + 18\), the parabolic curve opens downwards.

This is indicated by the negative coefficient in front of the \(x^2\) term, \(-0.5\). The vertex of the parabola is the highest or lowest point, depending on whether it opens upwards or downwards. In our sample function, the vertex is at \((6, 18)\), meaning it's the peak of the parabola.

Understanding the nature of parabolas helps to predict the path of objects in physics, like the trajectory of a jump. The symmetrical property of parabolas indicates that the path on both sides of the vertex is mirrored.

Vertex Form

The vertex form of a quadratic function is a convenient way to express the function while easily identifying the vertex of a parabola. It has the general format:\[f(x) = a(x - h)^2 + k,\] where \( (h, k) \) is the vertex of the parabola.

The vertex form shows the transformations applied to the function visually and helps identify the vertex quickly. For our pogo stick model \(f(x) = -0.5(x - 6)^2 + 18\), the vertex form reveals that the vertex is \((6, 18)\), and the parabola opens downwards as indicated by \(-0.5\).

By analyzing a function in vertex form, we can foresee real-world applications, like determining the maximum height of a jump. This form simplifies the process by readily providing information about the location and orientation of the parabola on the graph.