Question

Q8 Identify the key features of the graph below. Identify the y-intercept, the rate of decay/growth, and the common ratio. if it pass through (-2,1) and (0,3) and is an exponential growth function.

Answer 1

General exponential form: y = a·b^x.

Use the point (0,3): a = 3. So y = 3·b^x.

Use (-2,1): 1 = 3·b^(−2) ⇒ b^(−2) = 1/3 ⇒ b^2 = 3 ⇒ b = √3 (take the positive root for growth).

So the function is y = 3(√3)^x = 3·3^(x/2).

Key features:
- y-intercept: (0, 3).
- Common ratio (multiplier per +1 in x): b = √3 ≈ 1.732.
- Growth rate per unit x: b − 1 = √3 − 1 ≈ 0.732 ≈ 73.2% growth per unit.
- Since b > 1 it is exponential growth.
- Horizontal asymptote: y = 0; domain (−∞, ∞); range (0, ∞).

(Also note: over the 2-unit change from x = −2 to x = 0 the value multiplies by b^2 = 3, which matches 1 → 3.)