Problem 63 - Precalculus Essentials

Question

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$h(x)=2(x-2)^{2}-1$$

Step-by-Step Solution

Answer

The function \(h(x) = 2(x - 2)^2 - 1\) is a transformation of standard quadratic function shifted two units to the right, stretched vertically by a factor of 2, and shifted down by one unit. Its graph will be a narrower parabola, with vertex at (2,-1) and opening upwards.

Step 1: Understand the Transformations

Examine the given function \(h(x) = 2(x - 2)^2 - 1\) to understand the transformations from the standard function. \(x - 2\) in the brackets implies a 2 units shift to the right. Factor 2 multiplied with the squared function implies a vertical stretch of the graph by a factor of 2. Lastly, \(-1\) outside the brackets indicates a 1 unit downwards shift.

Step 2: Graph the Standard Function

Start with the standard function \(f(x) = x^2\). This is a parabola with vertex at the origin (0,0) and opens upwards. The shape of the graph is determined by plotting some key points such as (-2,4), (-1, 1), (0,0), (1,1), and (2,4). Draw the graph.

Step 3: Apply the Horizontal Shift

Shift each point on the standard graph 2 units to the right. This corresponds to the \(x - 2\) in the given function. The vertex will move from (0,0) to (2,0), and other points will adjust accordingly.

Step 4: Apply the Vertical Stretch

Stretch the graph vertically by a factor of 2. This corresponds to the factor '2' in the given function. Multiply the y-coordinate of each point by 2. For example, the vertex will remain at (2,0), while the point that was at (3,1) will now be at (3,2). The graph becomes narrower because of the vertical stretch.

Step 5: Apply the Vertical Shift

Shift the graph downwards by one unit. This corresponds to \(-1\) in the given function. It will move the vertex from (2,0) to (2,-1) and similarly all other points will be shifted down by 1 unit.

Step 6: Draw the Transformed Graph

Draw the final graph of the function \(h(x) = 2(x - 2)^2 - 1\), incorporating all the transformations done in the previous steps. The graph of \(h(x)\) will also be a parabola but will be narrower due to the vertical stretch, opening upwards, and its vertex will be at (2, -1).