Question

graph the equation shown below by transforming the given graph of the parent function.
$y = (x - 3)^2$

Explanation:

Step1: Identify the parent function

The parent function of \( y=(x - 3)^2 \) is \( y = x^2 \), whose graph is a parabola opening upwards with vertex at \((0,0)\).

Step2: Analyze the transformation

For a function of the form \( y=(x - h)^2 \), it represents a horizontal shift of the parent function \( y = x^2 \). If \( h>0 \), the graph shifts \( h \) units to the right. Here, \( h = 3 \), so the graph of \( y = x^2 \) (with vertex at \((0,0)\) and the given points) will shift 3 units to the right.

Step3: Determine the new vertex and points

• The vertex of \( y = x^2 \) is \((0,0)\). After shifting 3 units right, the new vertex is \((3,0)\).
• For the other points on \( y = x^2 \) (e.g., \((1,1)\), \((2,4)\), \((3,9)\) etc., and the symmetric points on the left), we add 3 to the \( x \)-coordinate of each point. For example, the point \((1,1)\) on \( y = x^2 \) will become \((1 + 3,1)=(4,1)\), the point \((2,4)\) will become \((2+3,4)=(5,4)\), the point \((3,9)\) will become \((3 + 3,9)=(6,9)\), and the symmetric points on the left like \((- 1,1)\) will become \((-1+3,1)=(2,1)\), \((-2,4)\) will become \((-2 + 3,4)=(1,4)\), \((-3,9)\) will become \((-3+3,9)=(0,9)\). Then we plot these new points and draw the parabola opening upwards with vertex at \((3,0)\).

Answer:

To graph \( y=(x - 3)^2 \), shift the graph of the parent function \( y = x^2 \) (given in the image) 3 units to the right. The new vertex is at \((3,0)\), and other points are obtained by adding 3 to the \( x \)-coordinates of the points on \( y = x^2 \). Then draw the parabola through these new points.