The sketch shows a curve with equation y=ab^x where a and b are constants and b>0.
The curve passes through the points (0,3) and (2,12).
Calculate the value of a and b.
Thank you.
sub in the two points to set up two equations
for (0,3)
3 = a b^0, but b^0 = 1
so a = 3
for (2,12)
12 = 3 b^2
4 = b^2
b = ± 2, but we need b> 0
a = 3 , b = 2 ---> y = 3 (2)^x
b^0 = 1, so a=3
Now you know that 3b^2 = 12
I expect you can find b now ...
Why did the mathematician bring a ladder to the equation? Because he wanted to find the "x"! But in all seriousness, let's solve this problem together.
We have two points that the curve passes through: (0,3) and (2,12). To find the values of a and b, we can substitute these values into the equation y = ab^x.
For the first point (0,3), we plug in x=0 and y=3:
3 = a * b^0
3 = a * 1
So we can determine that a = 3.
For the second point (2,12), we plug in x=2 and y=12:
12 = 3 * b^2
12 = 3 * b * b
Dividing both sides by 3, we get:
4 = b^2
Taking the square root of both sides, we find:
b = ±2
Since b must be greater than 0, we conclude that b = 2.
Therefore, the values of a and b are a = 3 and b = 2.
Hope this helps lighten up your math journey a bit! Let me know if you have any other questions.
To find the values of a and b, we will substitute the given points into the equation y = ab^x.
1. For the point (0,3):
Substitute x=0 and y=3 into the equation.
3 = a * b^0
3 = a * 1
Therefore, we have a = 3.
2. For the point (2,12):
Substitute x=2 and y=12 into the equation.
12 = 3 * b^2
Divide both sides by 3 to isolate b^2.
4 = b^2
Take the square root of both sides.
b = 2 (since b>0)
Thus, the values of a and b are a=3 and b=2, respectively.
To calculate the values of a and b, we need to use the information provided about the curve passing through the points (0,3) and (2,12).
Step 1: Substituting the coordinates (0,3) into the equation y = ab^x, we get:
3 = ab^0
Since any number raised to the power of 0 is 1, we can simplify the equation to:
3 = a
Step 2: Substituting the coordinates (2,12) into the equation y = ab^x, we get:
12 = ab^2
Since we know that a = 3 from the previous step, we can substitute it in the equation:
12 = 3b^2
Rewriting the equation:
4 = b^2
Taking the square root of both sides of the equation:
2 = b
So the values of a and b are:
a = 3
b = 2
Therefore, the equation of the curve is y = 3(2^x).