Question

use transformations of this graph to graph the given function. each function.

1. h(x) = (x - 6)² - 3

Explanation:

Step1: Identify Parent Function

The parent function is \( y = x^2 \), a parabola with vertex at \((0,0)\), opening upward.

Step2: Analyze Horizontal Shift

For \( h(x)=(x - 6)^2 - 3 \), the \( (x - 6) \) term means a horizontal shift. The rule for \( y = f(x - h) \) is a shift right by \( h \) units. Here, \( h = 6 \), so shift the parent parabola 6 units to the right. The vertex moves from \((0,0)\) to \((6,0)\).

Step3: Analyze Vertical Shift

The \( - 3 \) term means a vertical shift. The rule for \( y = f(x) - k \) is a shift down by \( k \) units. Here, \( k = 3 \), so shift the parabola (already shifted right) 3 units down. The vertex moves from \((6,0)\) to \((6, - 3)\).

Step4: Graph the Transformed Function

Using the transformed vertex \((6, - 3)\) and the same shape (opening upward, same width as \( y = x^2 \)), plot the parabola. Points on \( y = x^2 \) like \((1,1)\), \((-1,1)\) will transform: for \( x = 7 \) (6 + 1), \( h(7)=(7 - 6)^2 - 3=1 - 3=-2 \), so \((7, - 2)\); for \( x = 5 \) (6 - 1), \( h(5)=(5 - 6)^2 - 3=1 - 3=-2 \), so \((5, - 2)\), etc.

Answer:

To graph \( h(x)=(x - 6)^2 - 3 \), start with \( y = x^2 \), shift its vertex \((0,0)\) 6 units right to \((6,0)\), then 3 units down to \((6, - 3)\), and draw the upward - opening parabola through transformed points (e.g., \((5, - 2)\), \((7, - 2)\), \((6, - 3)\)). The vertex is at \((6, - 3)\), and the parabola has the same shape as \( y = x^2 \).