Problem 63 - 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. $$h(x)=2(x-2)^{2}-1$$

Step-by-Step Solution

Answer

The graph of \(h(x)=2(x-2)^{2}-1\) is a parabola that opens upwards with the vertex at (2,-1). It is derived from the graph of \(f(x)=x^{2}\) by shifting it 2 units to the right, stretching it vertically by a factor of 2 and shifting it 1 unit downwards.

Step 1: Graph the standard Quadratic Function

Start by sketching the graph of the standard quadratic function \(f(x)=x^{2}\). The graph is a U-shaped curve called a parabola, opening upwards, with a vertex at the origin (0,0).

Step 2: Apply Horizontal Shift

The function \(h(x)=2(x-2)^{2}-1\) involves a horizontal shift of 2 units to the right from the graph of \(f(x)=x^{2}\). So, every point on the \(f(x)=x^{2}\) graph will be moved 2 units to the right on the x-axis. For instance, the vertex of the parabola will move from (0,0) to (2,0).

Step 3: Apply Vertical Stretch

The number 2 in front of the squared term in \(h(x)=2(x-2)^{2}-1\), would cause the graph to stretch vertically by a factor of 2. In other words, the y-values for the quadratic function are doubled.

Step 4: Apply Vertical Shape

Finally, the '-1' at the end of \(h(x)=2(x-2)^{2}-1\) represents a vertical shift, which means it would bring every point on the graph down by 1 unit. Therefore, the vertex will finally move from (2,0) to (2,-1).

Step 5: Draw Final graph

With all the transformations applied, you can now sketch a final graph for \(h(x)=2(x-2)^{2}-1\), which is the graph of \(f(x)=x^{2}\) that is first horizontally shifted 2 units to right, then vertically stretched by a factor of 2 and lastly vertically shifted 1 unit down.