could someone explain this to me in an easy way to understand. Thank you kindly.
Point out that to graph the function
f (x) = a (x - h)^ 3+ k, you identify the point of
symmetry (h, k) and use the value of a to draw the
graph through two additional points
(-1 + h, –a + k) and (1 + h, a + k). So, to graph
f (x) = 2 (x - 3) ^3+ 1, first identify the point of
symmetry, (3, 1). Then identify points
(-1 + 3, -2 + 1) = (2, -1) , and
(1 + 3, 2 + 1) = (4, 3) as additional reference points.
A smooth curve through these three points is a good
beginning for the graph.
looks good to me
I mean, consider the graph of x^3
stretch it up by a factor of 2
shift it right by h and up by k
Since (-1,-1), (0,0), and (1,1) lie on the graph of x^3,
(-1+h,-a+k), (h,k), and (1+h,a+k) lie on the graph of f(x)
To graph the function f(x) = a(x - h)^3 + k, you can follow these steps:
1. Identify the point of symmetry: The point (h, k) represents the vertex of the graph, which is the point of symmetry. In the given function f(x) = 2(x - 3)^3 + 1, the point of symmetry is (3, 1).
2. Find two additional points: To draw the graph, you need to find two more points on the curve. You can do this by substituting x-values into the function.
- First additional point: Substitute x = -1 into the equation.
(-1 + h, -a + k) = (-1 + 3, -2 + 1) = (2, -1)
So, the first additional point is (2, -1).
- Second additional point: Substitute x = 1 into the equation.
(1 + h, a + k) = (1 + 3, 2 + 1) = (4, 3)
Therefore, the second additional point is (4, 3).
3. Plot the points: Plot the three points on a coordinate grid. Place a point at the vertex (3, 1), as well as at the two additional points (2, -1) and (4, 3).
4. Draw a smooth curve: Connect the three points with a smooth curve. The curve should pass through all three points. This curve represents the graph of the function f(x) = 2(x - 3)^3 + 1.
Remember that this is just a starting point for graphing the function. You can further refine the graph by plotting more points or using additional techniques.