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Chapter 2: Problem 57

Begin by graphing the standard quadratic function, \(f(x)=x^{2} .\) Then use transformations of this graph to graph the given function. $$h(x)=-(x-2)^{2}$$

Short Answer

Expert verified
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 by step solution

01

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).
02

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).
03

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.
04

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.

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