A increasing exponential function has the form f(x) = pqx + r
and f(x) satisfies the following conditions.
- f(x) > −3 ∀ x
- (0;−2) and (1;−1) are points on the graph
- Determine the equation of f
- f(x) is transformed to g(x) = 2.2x + 1 = 2x + 1 + 1 (Exponents)
Please define this transformation.
f(x) is an increasing exponential function
∴ q>1 and qx increases from 0 to ∞ ∀x ϶ −∞ < x < ∞
∴p > 0 so pqx > 0 ∴ pqx −3 > −3 ∴ let f(x) = pqx −3
∴ f(x) > −3 so the first condition is satisfied.
at (0;−2) f(0) = −2 ∴ pq⁰ −3 = −2 ∴ p = 1
∴ f(x) = qx −3
at ( 1;−1) f(1) = −1 ∴ q¹−3 = −1∴ q=2
so f(x) = 2x−3
∴ q>1 and qx increases from 0 to ∞ ∀x ϶ −∞ < x < ∞
∴p > 0 so pqx > 0 ∴ pqx −3 > −3 ∴ let f(x) = pqx −3
∴ f(x) > −3 so the first condition is satisfied.
at (0;−2) f(0) = −2 ∴ pq⁰ −3 = −2 ∴ p = 1
∴ f(x) = qx −3
at ( 1;−1) f(1) = −1 ∴ q¹−3 = −1∴ q=2
so f(x) = 2x−3
g(x) = 2x+1+1 ⇒ g(x) = 2x+1+1 −4+4 add zero
g(x) = 2x+1−3+4 ∴ g(x) = f(x+1) + 4
g(x) = 2x+1−3+4 ∴ g(x) = f(x+1) + 4
So for any (x,y) ≝ (x,f(x))
the transformation f(x+1) + 4
defines a translation one unit to the left
and 4 units up from (x,f(x)).
'≝' shall mean 'equal to by definition'.
Please check the 5 examples in the diagram.
the transformation f(x+1) + 4
defines a translation one unit to the left
and 4 units up from (x,f(x)).
'≝' shall mean 'equal to by definition'.
Please check the 5 examples in the diagram.