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)
shift or scale. Any ideas which go where?
i don't understand it can you
consider the graph
k(x) = x^2
k(x) = x^2+5 is the same graph, shifted up 5 units.
Similarly, k(x) = (x-5)^2 is the same graph, shifted to the right by 5 units.
If you think of the k-axis moved to the right 5 units, all the new x-coordinates are 5 less than the old ones. That's why substituting (x-5) is the same as moving the k-axis 5 units to the right.
Think of scaling the same way. The graph grows or shrinks because the x-coordinate grows or shrinks by a factor of c.
Visit http://rechneronline.de/function-graphs/ where you can play around with tweaking the functions and see how they are affected. You can display up to 3 graphs at once. So, enter
x^2 for the first,
(x-5)^2 for the 2nd, and
x^2+5 for the 3rd.
You might want to change x and y ranges from -5 to 5 and make them -10 to 10 instead. Just play around some.
To compare and contrast how the application of a constant, c, affects the graph of the function k(x) = x^2, we need to understand how each of the four different applications interacts with the function.
1. k(x + c):
When we apply a constant, c, to the variable x inside the function, we are shifting the graph horizontally (left or right) by c units. In this case, the graph of k(x + c) will be shifted c units to the left compared to the original graph of k(x). This means that the vertex of the parabola will be shifted c units to the left, while the shape of the graph remains the same.
2. k(x) + c:
Adding a constant, c, to the entire function value k(x) will shift the graph vertically (upwards or downwards) by c units. The entire graph of k(x) + c will be shifted c units upward compared to the original graph of k(x). This means that every point on the graph will be raised or lowered by c units while maintaining the same shape.
3. k(cx):
Applying a constant, c, to the variable x inside the function affects the stretch or compression of the graph. In this case, the graph of k(cx) will be horizontally compressed or stretched based on the value of c. If c > 1, the graph will be compressed horizontally, making it narrower, while if 0 < c < 1, the graph will be stretched horizontally, making it wider. The shape of the graph remains the same, but the x-values will be multiplied by c, impacting the width of the parabola.
4. c • k(x):
Multiplying the entire function by a constant, c, results in vertical stretching or compression. The graph of c • k(x) will be vertically stretched or compressed based on the value of c. If c > 1, the graph will be stretched vertically, making it taller, while if 0 < c < 1, the graph will be compressed vertically, making it shorter. The shape of the graph remains the same, but the y-values will be multiplied by c, influencing the height of the parabola.
To summarize, applying a constant to the variable x in k(x + c) shifts the graph horizontally, adding a constant to k(x) shifts the graph vertically, applying a constant to x in k(cx) compresses or stretches the graph horizontally, and multiplying k(x) by a constant stretches or compresses the graph vertically.