Let `y=f(x)=ab^x` represents the general form of the exponential function,
When a`>0` and b`>0` , b represents the growth factor,
When x=0 , then y=a which is the y-intercept.
2) Finding Grwoth factor and y-intercept from the table
Now let's have a sample table stated below to find out the y-intercept and growth factor.
If there is x value in the table having x=0 , then the corresponding y value represents the y-intercept, otherwise y-intercept...
Let `y=f(x)=ab^x` represents the general form of the exponential function,
When a`>0` and b`>0` , b represents the growth factor,
When x=0 , then y=a which is the y-intercept.
2) Finding Grwoth factor and y-intercept from the table
Now let's have a sample table stated below to find out the y-intercept and growth factor.
If there is x value in the table having x=0 , then the corresponding y value represents the y-intercept, otherwise y-intercept can be found as follows:
x y
1 6
2 12
3 24
4 48
Since the exponential function passes through the above points say (x_1,y_1) and (x_2,y_2), the points will satisfy the equation of the function,
`:.y_1=ab^(x_1)` -----equation 1
and `y_2=ab^(x_2)`
Dividing the above equations will yield,
`y_2/y_1=b^(x_2)/b^(x_1)`
`y_2/y_1=b^(x_2-x_1)`
or `b=(y_2/y_1)^(1/(x_2-x_1))`
Now from the table , plug in the values of the points to find the b,
`b=(12/6)^(1/(2-1))`
`b=2^(1/1)`
`b=2`
Now y-intercept (a) can be found by plugging in any of the equation,
Let's plug b in the equation 1
`y_1=ab^(x_1)`
`6=a(2)^1`
`6=2a`
`a=6/2`
`a=3`
Therefore y-intercept=3
3) Finding y-intercept and growth rate from the graph
Pl see the attached graph.
Look at the graph, y-intercept will be the y-coordinate where the graph of the function intersects the y-axis.
From the graph y-intercept=3
Growth factor can be found by noting the x and y-coordinates and then plugging them in the equation as follows:
From the graph,
`x_2=2,y_2=12`
`x_1=1,y_1=6`
So growth factor=`(y_2/y_1)^(1/(x_2-x_1))`
`b=(12/6)^(1/(2-1))`
`b=2^1`
b=2
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