Problem 66 - 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 the given function \(h(x) = -2(x+2)^{2} + 1\) is obtained from the standard quadratic function by a vertical stretch by a factor of -2, shifting 2 units to the left and shifting 1 unit upwards.

Step 1: Graph the standard Quadratic Function

Begin by graphing the standard quadratic function \(f(x) = x^{2}\). This is a parabola opening upwards with the vertex at the origin (0,0).

Step 2: Apply the Vertical Stretch

The coefficient -2 in the function \(h(x)\) represents a vertical stretch by a factor of 2, reflecting the graph across the x-axis. This makes the graph open downwards.

Step 3: Apply the Horizontal Shift

The \(x+2\) in the function indicates a horizontal shift of the graph. The sign is opposite to the actual direction of the shift, hence the graph shifts 2 units to the left.

Step 4: Apply the Vertical Shift

The term +1 at the end of the function indicates a vertical shift. As the sign is positive, the graph moves upwards by 1 unit.

Step 5: Plot the transformed function

After applying all the transformations to the standard quadratic function, graph the transformed function \(h(x) = -2(x+2)^{2} + 1\).