Problem 57 - 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. $$h(x)=-(x-2)^{2}$$

Step-by-Step Solution

Answer

The function \(h(x) = -(x-2)^{2}\) presents as a parabola shifted 2 units to the right, compared to \(f(x) = x^{2}\), and is reflected over the x-axis.

Step 1: Understanding the Base Function

The standard quadratic function is \(f(x) = x^2\). Its graph is a parabola that opens upwards with the vertex at the origin (0,0).

Step 2: Applying the Shift

The function \(h(x) = -(x-2)^{2}\) is the basic quadratic function shifted 2 units to the right. You achieve this shift by replacing every \(x\) in the original function with \((x - 2)\). Now the vertex of the parabola is at (2, 0).

Step 3: Applying the Reflection

The negative sign in front of the square implies a reflection about the X-axis. This means that the parabola, which opened upwards in the original function, now opens downwards.

Step 4: Drawing the Transformed Function

Based on the transformations outlined in the previous steps, draw the transformed graph. Start at the vertex (2, 0) and draw a parabola opening downward.