Problem 65 - 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. $$h(x)=-2(x+1)^{2}+1$$

Step-by-Step Solution

Answer

The graph of \(h(x)=-2(x+1)^{2}+1\) would be a downward facing parabola that is stretched by a factor of 2 and shifted 1 unit to the left and 1 unit upward from the standard graph of \(f(x)=x^{2}\).

Step 1: Graph the Standard Quadratic Function

Start by plotting the graph of standard quadratic function \(f(x)=x^{2}\). It is a symmetrical U-shape graph with the vertex at the origin (0,0) and the axis of symmetry along the y-axis.

Step 2: Identify the Transformations

Now, looking at the function \(h(x)=-2(x+1)^{2}+1\), it is a quadratic function just like \(f(x)=x^{2}\) but with some transformations applied. More specifically, there is a horizontal shift, a vertical shift and a vertical stretch and flip. The shift to the left is due to \(x+1\), that is a shift 1 unit to the left. The term \(+1\) at the end implies a vertical shift of 1 unit upwards. The coefficient -2 implies a vertical stretch by a factor of 2 and a flip over the x-axis.

Step 3: Plot the Transformed Graph

Apply these transformations to the standard graph of \(f(x)=x^{2}\). The entire graph moves 1 unit to the left because of the \(x+1\), it then moves 1 unit upward because of the \(+1\) and finally, the graph is vertically stretched by a factor of 2 and then flipped over the x-axis due to the presence of the -2 coefficient. After these transformations, the graph of \(h(x)=-2(x+1)^{2}+1\) is obtained.