Given the function k(x) = x2, compare and contrast how the application of a constant, c, affects the graph. The application of the constant must be discussed in the following manners:
k(x + c)
k(x) + c
k(cx)
c • k(x)
The original function is k(x) = x^2. Provided that c cannot be equal to zero, we'll compare the following to the original position of the given function:
1. k(x+c) = (x+c)^2
The graph moves c units to the left is c is (+) and c units to the right if c is (-).
2. k(x) + c = x^2
The graph moves c units down when c is (+) and c units up when c is (-).
3. k(cx) = (cx)^2
Regardless of the sign, the graph becomes more narrow compared to the original if c is less than -1 and greater than 1, and becomes wider compared to the original if c is between -1 and 1 (if c is equal to zero, the graph becomes a line with equation y = 0).
By the way I can't understand the last (the c • k(x)). ^^;
Hope this helps~ :3
Given the function k(x) =x^2, compare and contrast how the application of a constant,c, affects the graph. The application of the constant must be discussed in the following manners: k(cx) c times k(x).
To compare and contrast how the application of a constant, c, affects the graph of the function k(x) = x^2, let's discuss each case:
1. k(x + c): When you apply a constant c to the input of the function by adding it to x, it horizontally shifts the graph of the function to the left or right. If c > 0, the graph shifts c units to the left, and if c < 0, the graph shifts |c| units to the right. This is because adding c to x affects all points on the graph by the same value.
2. k(x) + c: In this case, the constant c is added to the output of the function. This vertically shifts the graph of the function up or down. If c > 0, the graph shifts c units upward, and if c < 0, the graph shifts |c| units downward. The value of c affects the y-coordinate of each point on the graph, causing the shift.
3. k(cx): Here, the constant c is multiplied by x before evaluating the function. This stretches or compresses the graph horizontally. If |c| > 1, the graph is compressed horizontally towards the y-axis. If |c| < 1, the graph is stretched horizontally away from the y-axis. The closer c is to 0, the closer the graph becomes to a vertical line.
4. c • k(x): In this case, the constant c is multiplied by the whole function. This results in a vertical stretching or compressing of the graph. If |c| > 1, the graph is stretched vertically. If |c| < 1, the graph is compressed vertically. The value of c multiplies the y-coordinate of each point on the graph, hence the vertical scaling.
In summary:
- Applying a constant to the input of the function (x + c) shifts the graph horizontally.
- Adding a constant to the output of the function (k(x) + c) shifts the graph vertically.
- Multiplying the input of the function by a constant (k(cx)) stretches or compresses the graph horizontally.
- Multiplying the output of the function by a constant (c • k(x)) stretches or compresses the graph vertically.