Question

what is the equation of the function shown in the graph, given the equation of the parent function is $f(x) = 3.5^x$?
\\(\circ\\) $g(x) = 3.5^x + 1.5$
\\(\circ\\) $g(x) = 3.5^x - 2$
\\(\circ\\) $g(x) = 3.5^x - 1$
\\(\circ\\) $g(x) = 3.5^x + 1$
(graph: a coordinate plane with a curve labeled $g(x)$, passing through $(0, -1)$ and $(1, 1.5)$, with a horizontal asymptote at $y = -2$.)

Explanation:

Step1: Use the point (0, -1)

Substitute \( x = 0 \) and \( y = -1 \) into each option.

• For \( g(x)=3.5^x + 1.5 \): \( 3.5^0 + 1.5 = 1 + 1.5 = 2.5

eq -1 \)

• For \( g(x)=3.5^x - 2 \): \( 3.5^0 - 2 = 1 - 2 = -1 \) (holds for (0, -1))
• For \( g(x)=3.5^x - 1 \): \( 3.5^0 - 1 = 1 - 1 = 0

eq -1 \)

• For \( g(x)=3.5^x + 1 \): \( 3.5^0 + 1 = 1 + 1 = 2

eq -1 \)

Step2: Verify with point (1, 1.5)

Substitute \( x = 1 \) into \( g(x)=3.5^x - 2 \): \( 3.5^1 - 2 = 3.5 - 2 = 1.5 \), which matches the point (1, 1.5).

Answer:

\( g(x) = 3.5^x - 2 \) (the second option: \( g(x) = 3.5^x - 2 \))