what do the a, h, and k values do in the graphs of these functions??
f(x) = a*log(x-h)+k
f(x) = a*2^(x-h) +k
i think the k is the horizontal asymptope but i dont know about the others
k moves the graphs up a distance k
h moves the graphs left a distance h
a makes the graphs steeper
f(x-h) moves to the right by h
a makes the graph steeper if |a| > 1; flatter if |a| < 1
a flips the graph upside down if a < 0
In the functions f(x) = a*log(x-h) + k and f(x) = a*2^(x-h) + k, the values of a, h, and k affect different aspects of the graphs.
For the function f(x) = a*log(x-h) + k:
1. The value of "a" determines the vertical stretch or compression of the graph. If "a" is positive, the graph will be stretched vertically. If "a" is negative, the graph will be compressed vertically and reflected across the x-axis.
2. The value of "h" represents the horizontal translation of the graph. It determines the location of the vertical asymptote and the point where the graph is shifted horizontally. If "h" is positive, the graph will shift to the right, while if "h" is negative, the graph will shift to the left.
3. The value of "k" controls the vertical translation or shift of the graph. It moves the entire graph up or down along the y-axis. If "k" is positive, the graph will shift upward, and if "k" is negative, the graph will shift downward. "k" also represents the y-coordinate of the horizontal asymptote.
For the function f(x) = a*2^(x-h) + k:
1. The value of "a" determines the vertical stretch or compression of the graph. If "a" is positive, the graph will be stretched vertically. If "a" is negative, the graph will be compressed vertically and reflected across the x-axis.
2. The value of "h" represents the horizontal translation of the graph. Similar to the previous function, it determines the location of the point on the graph when it is shifted horizontally. If "h" is positive, the graph will shift to the right, while if "h" is negative, the graph will shift to the left.
3. The value of "k" controls the vertical translation or shift of the graph. It moves the entire graph up or down along the y-axis. If "k" is positive, the graph will shift upward, and if "k" is negative, the graph will shift downward.
In the given functions, f(x) = a*log(x-h) + k and f(x) = a*2^(x-h) + k, the variables a, h, and k represent different characteristics of the graphs of these functions.
1. a (amplitude/vertical stretch):
The parameter 'a' is the amplitude or vertical stretch factor of the function. In the logarithmic function f(x) = a*log(x-h) + k, 'a' determines how much the graph is scaled vertically. If 'a' is positive, the graph will stretch vertically by a factor of 'a'. Conversely, if 'a' is negative, the graph will flip vertically. In the exponential function f(x) = a*2^(x-h) + k, 'a' also represents the vertical stretch, with similar properties as described above.
2. h (horizontal shift):
The parameter 'h' represents the horizontal shift or translation of the function. In both logarithmic and exponential functions, 'h' indicates the number of units the graph is shifted horizontally. If 'h' is positive, the graph will shift to the right by 'h' units. On the other hand, if 'h' is negative, the graph will shift to the left by 'h' units.
3. k (vertical shift):
The parameter 'k' indicates the vertical shift or translation of the function. In both functions, 'k' determines the number of units the graph is shifted vertically. If 'k' is positive, the graph will shift upward by 'k' units. Conversely, if 'k' is negative, the graph will shift downward by 'k' units.
It's worth noting that in both cases, neither 'k' nor 'a' affect horizontal asymptotes. Horizontal asymptotes are determined solely by the behavior of the functions as x approaches positive or negative infinity.