Answered

write the equation for an exponential function, in the forms y = a x b^x, whose graph passes through the coordinate points (1, 7.5) and (3, 16.875).

Verified Answer :

Panoyin

y = abˣ
(a ≠ 0, b ≠ 0)

(1, 7.5)
x = 1
y = 7.5
    ↓
7.5 = ab

(3, 16.875)
x = 3
y = 16.875
        ↓
16.875 = ab³

   7.5 = ab
     b      b
7.5/b = a

    16.875 = ab³
         b³        b³
16.875/b³ = a

     7.5/b  = 16.875/b³
7.5/b(b³) = 16.875/b³(b³)
     7.5b² = 16.875
      7.5         7.5
          b² = 2.25
      √(b²) = √(2.25)
           b = 1.5

a = 7.5/b
a = 7.5/1.5
a = 5

y = 5(1.5)ˣ

Answer:

The the required exponential function is \(y=5(1.5)^x\).

Step-by-step explanation:

The equation for an exponential function, in the forms

\(y=ab^x\)

It is given that passes through the coordinate points (1, 7.5) and (3, 16.875). It means the given function must be satisfied by these points.

\(7.5=ab^1\)

\(7.5=ab\)                        ....(1)

\(16.875=ab^3\)              ..... (2)

Divide equation (2) by equation (1).

\(\frac{16.875}{7.5}={ab^3}{ab}\)

\(2.25=b^2\)

\(b=1.5\)

The value of b is 1.5. Substitute b=1.5 in equation (1).

\(7.5=a(1.5)\)

Divide both sides by 1.5.

\(a=5\)

Therefore the required exponential function is \(y=5(1.5)^x\).