News
Get Premium
Dashboard

Dashboard Sign out
Core Connections Algebra 1, 2013
CC
Core Connections Algebra 1, 2013 View details
2. Section 7.2
Continue to next subchapter
search

Exercise 96 Page 346

Practice makes perfect
a We want to write an exponential function for the graph that passes through the given points. Let's consider the general form for this type of function.
y=ab^x Since we want the points to lie on the graph, they must satisfy this equation. After substituting both points into the above formula, we obtain a system of equations.

7.5=a * b^1 & (I) 16.875=a * b^3 & (II) We want to solve this system of equations using substitution. There are three steps to follow.

  1. Isolate a variable in one of the equations.
  2. Substitute the expression for that variable into the other equation and solve.
  3. Substitute this solution into one of the equations and solve for the value of the other variable.
Observing the given equations, it looks like it will be simplest to isolate a in the first equation.
7.5=a * b^1 & (I) 16.875=a * b^3 & (II)
▼
(I): Solve for a
ExponentOne

(I): a^1=a

7.5=a * b 16.875=a * b^3
DivEqn

(I): .LHS /b.=.RHS /b.

7.5b=a 16.875=a * b^3
RearrangeEqn

(I): Rearrange equation

a= 7.5b 16.875=a * b^3
Now that we have isolated a we can solve the system by substitution.
a= 7.5b 16.875=a * b^3
Substitute

(II):a= 7.5/b

a= 7.5b 16.875= 7.5b * b^3
SplitIntoFactors

(II): Split into factors

a= 7.5b 16.875= 7.5b * b * b^2
FracMultDenomToNumber

(II): a/b* b = a

a= 7.5b 16.875=7.5 * b^2
▼
(II): Solve for b
DivEqn

(II): .LHS /7.5.=.RHS /7.5.

a= 7.5b 2.25=b^2
SqrtEqn

(II): sqrt(LHS)=sqrt(RHS)

a= 7.5b 1.5=b
RearrangeEqn

(II): Rearrange equation

a= 7.5b b=1.5
Great! Now, to find the value of a we need to substitute b=1.5 into either one of the equations in the given system. Let's use the first equation.
a= 7.5b b=1.5
Substitute

(I):b= 1.5

a= 7.5 1.5 b=1.5
CalcQuot

(I): Calculate quotient

a=5 b=1.5
The solution to this system of equations are the values a= 5 and b= 1.5. With this information, we can write the full equation of the exponential function. y= a b^x ⇒ y= 5( 1.5)^x
b We want to write an exponential function for the graph that passes through the given points. Let's consider the general form for this type of function.
y=ab^x Since we want the points to lie on the graph, they must satisfy this equation. After substituting both points into the above formula, we obtain a system of equations.

1.25=a * b^(- 1) & (I) 0.032=a * b^3 & (II) We want to solve this system of equations using substitution. There are three steps to follow.

  1. Isolate a variable in one of the equations.
  2. Substitute the expression for that variable into the other equation and solve.
  3. Substitute this solution into one of the equations and solve for the value of the other variable.
Observing the given equations, it looks like it will be simplest to isolate a in the first equation.
1.25=a * b^(- 1) & (I) 0.032=a * b^3 & (II)
▼
(I): Solve for a
NegOneExponentToFrac

(I): a^(- 1)=1/a

1.25=a * 1b 0.032=a * b^3
MultEqn

(I): LHS * b=RHS* b

1.25b=a 0.032=a * b^3
RearrangeEqn

(I): Rearrange equation

a=1.25b 0.032=a * b^3
Now that we have isolated a we can solve the system by substitution.
a=1.25b 0.032=a * b^3
Substitute

(II):a= 1.25b

a=1.25b 0.032= 1.25b * b^3
MultPow

(II): a^m*a^n=a^(m+n)

a=1.25b 0.032=1.25 * b^4
▼
(II): Solve for b
DivEqn

(II): .LHS /1.25.=.RHS /1.25.

a=1.25b 0.0256=b^4
RadicalEqn

(II): sqrt(LHS)=sqrt(RHS)

a=1.25b sqrt(0.0256)=b
CalcRoot

(II): Calculate root

a=1.25b 0.4=b
RearrangeEqn

(II): Rearrange equation

a=1.25b b=0.4
Great! Now, to find the value of a we need to substitute b=0.4 into either one of the equations in the given system. Let's use the first equation.
a=1.25b b=0.4
Substitute

(I):b= 0.4

a=1.25( 0.4) b=0.4
Multiply

(I): Multiply

a=0.5 b=0.4
The solution to this system of equations are values a= 0.5 and b= 0.4. With this information, we can write the full equation of the exponential function. y= a b^x ⇒ y= 0.5( 0.4)^x
Subchapter links expand_more
arrow_right 7.2.1 Curve Fitting and Fractional Exponents p.344
87
Curve Fitting and Fractional Exponents 88 (Page 344)
Curve Fitting and Fractional Exponents 89 (Page 344)
90
Curve Fitting and Fractional Exponents 91 (Page 344)
arrow_right 7.2.2 More Curve Fitting p.346-347
More Curve Fitting 96 (Page 346)
97
More Curve Fitting 98 (Page 347)
More Curve Fitting 99 (Page 347)
100
More Curve Fitting 101 (Page 347)
arrow_right 7.2.3 Solving a System of Exponential Functions Graphically p.352-353
Solving a System of Exponential Functions Graphically 106 (Page 352)
107
Solving a System of Exponential Functions Graphically 108 (Page 352)
109
Solving a System of Exponential Functions Graphically 110 (Page 353)
Solving a System of Exponential Functions Graphically 111 (Page 353)