Problem 66 - 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

In the end, the graph of \(h(x)=-2(x+2)^{2}+1\) will be a vertically reflected, compressed parabola shifted 2 units to the left and 1 unit up with the vertex at point (-2,1).

Step 1: Draw the base function

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

Step 2: Apply horizontal transformation

In \(h(x)= -2(x+2)^{2}+1\), (x+2) shifts the graph 2 units to the left. Now, the vertex of the parabola is at (-2,0).

Step 3: Apply vertical transformations

First apply the vertical compression by a factor of 2, making the parabola narrower. Then, the coefficient -2 applies vertical reflection, flipping the parabola over the x-axis. Now, the parabola opens downwards.

Step 4: Apply vertical shift

The +1 in the equation moves the graph 1 unit upward. Now, the vertex point of the parabola is at (-2,1).

Step 5: Draw the final graph

After performing all transformations, draw the final graph. The parabola opens downwards, it's narrower compared to the original function, and the vertex is located at (-2,1).