Problem 61 - College Algebra

Question

Begin by graphing the standard quadratic function, \(f(x)-x^{2} .\) Then use transformations of this graph to graph the given function. $$ g(x)-2(x-2)^{2} $$

Step-by-Step Solution

Answer

The graph of the given function \(g(x) = 2(x-2)^2\) is a parabola that opens upwards. This graph is obtained from the graph of the standard quadratic function \(f(x) = x^2\) by applying a vertical stretch (by a factor of 2), and a horizontal shift to the right (by 2 units).

Step 1: Graph the standard quadratic function

Start by graphing the standard quadratic function, \(f(x) = x^2\). This will serve as a base for further transformations.

Step 2: Reflect the graph

The standard quadratic function has a vertex at the origin (0,0). Drawing this will help in visualizing the transformation.

Step 3: Apply the vertical stretch

Now, to transform \(f(x) = x^2\) into \(g(x) = 2(x-2)^2\), start with the vertical stretch. This means that every y-coordinate will be twice as far from the x-axis. In other words, every point (x, y) on the graph of \(f(x) = x^2\) will be transformed to the point (x, 2y) on the graph of g(x). \nDraw this new graph.

Step 4: Apply the horizontal shift

Following the vertical stretch, apply the horizontal shift. This will move the graph 2 units to the right. That is, each point (x, y) on the graph just drawn will be transformed to the point (x+2, y) on the final graph of \(g(x) = 2(x-2)^2\). So, redraw the graph incorporating this horizontal shift.\nThis is the graph of \(g(x) = 2(x-2)^2\).

Key Concepts

Quadratic Function

A quadratic function is a type of polynomial function characterized by a degree of 2. It follows the general form of \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a eq 0\).
The graph of a quadratic function is a curve called a parabola, which is symmetric along a vertical line called the axis of symmetry. This line runs through the graph at the point where the parabola has its minimum or maximum value, known as the vertex.
For the function \(f(x) = x^2\), this is a simple parabola with its vertex at the origin \((0,0)\) and opens upwards. This standard form serves as the starting point for understanding how transformations can adjust the graph of a quadratic function.

Graph Transformations

Graph transformations are techniques used to modify the shape and position of a graph. They include operations such as translations, stretches, reflections, and more, allowing us to manipulate how a function's graph appears in a coordinate plane.
By applying transformations, we are able to recognize and visualize changes in a graph without having to plot numerous points individually, helping us understand relationships between different function forms.
For example, in altering a quadratic function from \(f(x) = x^2\) to \(g(x) = 2(x-2)^2\), transformations like vertical stretch and horizontal shift are essential steps in graphing the new function.

Vertical Stretch

A vertical stretch involves enlarging the distance of every point on the graph vertically from the x-axis. Essentially, this transformation makes the graph taller or steeper.
When applied to a function, a vertical stretch is performed by multiplying the y-coordinates by a factor. If the factor is greater than 1, the stretch makes the function steeper. In our exercise, transforming the graph of \(f(x) = x^2\) to \(g(x) = 2(x-2)^2\), involves stretching by a factor of 2, transforming each point \((x, y)\) on \(f(x)\) to \((x, 2y)\).
Such a transformation intensifies the slope of the quadratic curve, resulting in points moving away from each other along the vertical axis.

Horizontal Shift

A horizontal shift relocates the entire graph to the right or left along the x-axis. In this transformation, the x-coordinates of every point on the graph are adjusted while the y-coordinates remain unchanged.
For the quadratic function in the exercise, after the vertical stretch, we perform a horizontal shift to obtain \(g(x) = 2(x-2)^2\).
This shift is executed by altering \(x\) to \(x-2\), effectively moving the graph 2 units to the right, so that every (x,y) point becomes \((x+2, y)\).
Horizontal shifts are an effective way to adjust the position of a graph without changing its shape or orientation, helping graphers to set their functions in desired locations on the plane.