Chapter Closure
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Start by using the Slope Formula to find the slope.
Create a system of equations by substituting the given points into the equation of an exponential function, y=ab^x.
y=- 8x+10
y=6(1/3)^x
An equation in slope-intercept form follows a specific format.
y= mx+b
A slope of -8 means that for every 1 horizontal step in the positive direction we take 8 vertical steps in the negative direction. Now that we know the slope, we can write a partial version of the equation. y= -8 x+b To complete the equation we also need to determine the y-intercept, b. Since we know that the given points will satisfy the equation, we can substitute one of them into the equation to solve for b. Let's use ( 1, 2).
x= 1, y= 2
Identity Property of Multiplication
LHS+8=RHS+8
Rearrange equation
A y-intercept of 10 means that the line crosses the y-axis at the point (0,10). We can now complete the equation. y= - 8x+10
We want to find an exponential function that passes through (- 1, 18) and (1,2). To do this, we will create a system of equations by substituting ( x, y) = ( - 1, 18) into the general form of exponential function, y=ab^x, and then substituting again with ( x, y) = ( 1, 2).
18=a * b^(- 1) & (I) 2=a * b^1 & (II)
We want to solve the above system of equations.
When solving a system of equations using substitution, there are three steps.
Observing the given equations, it looks like it will be simplest to isolate a in the first equation.
(II): a^1=a
(I): a^(- 1)=1/a
(I): LHS * b=RHS* b
(I): Rearrange equation
Now that we have isolated a we can solve the system by substitution.
(II):a= 18b
(II): a* a=a^2
(II): .LHS /18.=.RHS /18.
(II): a/b=.a /2./.b /2.
(II): sqrt(LHS)=sqrt(RHS)
(II): Calculate root
(II): Rearrange equation
Great! Now, to find the value of a we need to substitute b= 13 into either one of the equations in the given system. Let's use the first equation.
(I):b= 13
(I): a* 1/b= a/b
(I): Calculate quotient
The solution to this system of equations is a=6 and b=13. With this information, we can finally write an equation of the exponential function that passes through the given points. y=ab^x ⇔ y=6 (1/3)^x