Problem 58 - 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. $$h(x)=-(x-1)^{2}$$

Step-by-Step Solution

Answer

The graph of the function \(h(x)=-(x-1)^{2}\) is a parabola opening downwards with vertex at the point (1,0). This is achieved by reflecting the graph of \(f(x) = x^{2}\) over the x-axis and shifting it 1 unit to the right.

Step 1: Graph the Standard Quadratic Function.

Start by drawing the graph of the standard quadratic function \(f(x) = x^{2}\). The graph is a parabola that opens upwards with vertex at the origin (0,0). It is symmetric about the y-axis.

Step 2: Identify and Understand the Effect of Each Transformation.

The given function \(h(x)=-(x-1)^{2}\) is the result of applying two transformations to the graph of \(f(x)=x^{2}\). - The negative sign outside the parenthesis flips the graph over the x-axis (it makes the parabola open downwards instead of upwards).- The (x-1) inside the parentheses shifts the graph 1 unit to the right.

Step 3: Apply the Transformations to the Original Graph.

Begin with the graph of \(f(x) = x^{2}\) which opens upwards and shifts it 1 unit to the right to match the (x-1) in the function. Then flip it over the x-axis to account for the negative sign on the outside of the parentheses. The resulting graph should open downwards with the vertex at the point (1,0).