Problem 61 - College Algebra Essentials

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 function \(g(x)=2(x-2)^{2}\) is a parabola that opens upward, narrower than the standard quadratic function, shifted 2 units to the right.

Step 1: Graph the Standard Quadratic Function

Start by drawing the graph of the standard quadratic function \(f(x) = x^2\). This is a parabola that opens upwards, with its vertex at the origin (0,0).

Step 2: Identify the Transformations

The given function \(g(x)=2(x-2)^{2}\) represents a transformation of the standard quadratic function. The number 2 in front of the quadratic indicates a vertical stretch by a factor of 2. The term \((x-2)\) in the quadratic indicates a horizontal shift 2 units to the right.

Step 3: Apply the Vertical Stretch

The vertical stretch by a factor of 2 means that every y-value of the original function \(f(x) = x^2\) will be doubled in the new function. Multiply the y-coordinate of each point on \(f(x) = x^2\) by 2 to create the stretched graph.

Step 4: Apply the Horizontal Shift

A horizontal shift to the right by 2 units means that to every x-value of the original function \(f(x) = x^2\) we will add 2. So, take every point on the graph from Step 3 and shift it 2 units to the right to get the final graph of \(g(x)=2(x-2)^{2}\).

Step 5: Combine Transformations and Graph \(g(x)\)

Apply both transformations to the graph of \(f(x) = x^2\) and generate the graph of \(g(x)=2(x-2)^{2}\). The final graph should still be a parabola that opens upward, but it would be narrower due to the vertical stretch, and the vertex of the parabola will be at the point (2,0) due to the horizontal shift.