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

Step-by-Step Solution

Answer

For the given function \( h(x)=-(x-1)^{2} \), graph shows a parabola, originally at the origin, that has been shifted 1 unit to the right and reflected over the x-axis.

Step 1: Graphing The Standard Quadratic Function

Start by plotting \( f(x)=x^{2} \). This is a basic quadratic function which forms a parabola opening upwards. The vertex or the minimum point of this parabola is at the origin (0,0). For a better understanding of the function behavior, points on either side of the vertex can be plotted. Typically, some easy-to-calculate points such as (-2,4), (-1,1), (0,0), (1,1), and (2,4) are chosen.

Step 2: Identify The Transformation Of The Given Function

The given function is \( h(x)=-(x-1)^{2} \). When comparing this function to the standard quadratic form, \( ax^2+bx+c \), it's clear that there are transformations. Specifically, the transformations include a horizontal shift to the right by 1 unit and a vertical reflection over the x-axis.

Step 3: Applying Transformations To The Graph

Applying the identified transformations to the original graph, the parabola is moved 1 unit to the right (the vertex becomes (1,0)) and then reflected over the x-axis, as the negative sign indicates.