Problem 59 - 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. $$h(x)=(x-2)^{2}+1$$

Step-by-Step Solution

Answer

The graph of the function \( h(x) = (x-2)^2 + 1 \) is the graph of the base function \( f(x) = x^2 \), shifted 2 units to the right and then 1 unit upward.

Step 1: Graph the base function

Let's start by drawing the graph of the basic quadratic function which is \( f(x) = x^2 \).

Step 2: Transform by moving horizontally

To transform the function \( f(x) = x^2 \) into \( h(x) = (x-2)^2 + 1 \), we start by shifting all points of the graph of \( f(x) = x^2 \) to the right by 2 units. This action is resulting from the effect of \( (x-2)^2 \) compared to \( x^2 \). So, now the function became \( (x-2)^2 \).

Step 3: Transform by moving vertically

Now, after moving horizontally, we need to shift the graph upwards by 1 unit. This is because we have a '+1' outside the squared term in \( h(x) = (x-2)^2 + 1 \). Therefore, the graph of the function \( h(x) = (x-2)^2 \) is pushed up by one unit, giving us the final function \( (x-2)^2 + 1 \).