Question

sketch the graph of the function.
$f(x)=4(x - 6)^{2}+1$

use the graphing tool to graph the function.
click to enlarge graph

Explanation:

Step1: Identify the vertex form

The function \( f(x) = 4(x - 6)^2 + 1 \) is in vertex form \( f(x)=a(x - h)^2 + k \), where \( (h,k) \) is the vertex. Here, \( h = 6 \), \( k = 1 \), and \( a = 4 \).

Step2: Determine the direction and vertex

Since \( a = 4>0 \), the parabola opens upward. The vertex is at \( (6,1) \).

Step3: Find the axis of symmetry

The axis of symmetry is the vertical line \( x = h \), so \( x = 6 \).

Step4: Find a few more points

• When \( x = 5 \), \( f(5)=4(5 - 6)^2 + 1 = 4(1)+1 = 5 \), so the point is \( (5,5) \).
• When \( x = 7 \), \( f(7)=4(7 - 6)^2 + 1 = 4(1)+1 = 5 \), so the point is \( (7,5) \).
• When \( x = 4 \), \( f(4)=4(4 - 6)^2 + 1 = 4(4)+1 = 17 \), so the point is \( (4,17) \).
• When \( x = 8 \), \( f(8)=4(8 - 6)^2 + 1 = 4(4)+1 = 17 \), so the point is \( (8,17) \).

Using these points (vertex, axis of symmetry, and other points), we can sketch the parabola. The graph will have its vertex at \( (6,1) \), open upward, and pass through the points calculated above.

Answer:

The graph is a parabola opening upward with vertex at \((6, 1)\), axis of symmetry \(x = 6\), and passing through points like \((5, 5)\), \((7, 5)\), \((4, 17)\), \((8, 17)\) (and can be sketched using these features).