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

Step-by-Step Solution

Answer

After applying all the transformations: horizontal shift (1 unit to the left), vertical reflection (due to the negative sign), vertical scaling (due to the factor of 2), and vertical shift (1 unit up), the graph of the given function \(h(x) = -2(x+1)^2 + 1\) can be drawn. The vertex of the transformed function is at (-1, 1).

Step 1: Graph the standard quadratic function

Start by drawing the graph of the standard quadratic function \(f(x) = x^2\). This is a parabolic curve that opens upward with the vertex at the origin (0,0).

Step 2: Apply horizontal translation

The term '(x+1)' within the function \(h(x) = -2(x+1)^2 + 1\) introduces a horizontal translation of the graph. Due to the +1 inside the parentheses, the graph of the original function will shift to the left by 1 unit.

Step 3: Apply vertical scaling

The factor -2 which multiplies the term \((x+1)^2\), introduces a vertical scaling and a vertical reflection of the graph. The negative sign means that the graph will be reflected about the x-axis, i.e., the original parabola, which opens upwards, will now open downwards. And the factor 2 means the graph will be vertically 'stretched', i.e., y values will be multiplied by 2 (increasing the steepness of the parabola).

Step 4: Apply vertical translation

The '+1' term outside the parentheses in the function \(h(x) = -2(x+1)^2 + 1\) brings a vertical translation. The entire graph is shifted up by 1 unit.

Step 5: Plot the transformed function

Finally, combine all the transformations - horizontal translation (1 unit to the left), vertical reflection (due to the negative sign), vertical scaling (due to the factor of 2) and vertical translation (1 unit up) - to graph the given function \(h(x) = -2(x+1)^2 + 1\). The vertex of this new, transformed parabola will be at the point (-1, 1).