Graphs of three exponential functions f, g, and h, illustrating their growth rates based on different bases.

Let’s solve this step by step.

We are given three exponential functions:

- f(x) = 100 · 3^x
- g(x) = 100 · (3.5)^x
- h(x) = 100 · 4^x

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Question 1: Which grows least quickly? Which grows most quickly? Explain.

All three functions have the same starting value (when x=0, they all equal 100), but they grow at different rates because of their bases: 3, 3.5, and 4.

In exponential functions like y = a·b^x, if b > 1, then the larger the base b, the faster the function grows as x increases.

So:
- Base 3 → slowest growth
- Base 3.5 → medium growth
- Base 4 → fastest growth

Therefore:
→ f(x) grows the least quickly (base 3).
→ h(x) grows the most quickly (base 4).

You can test this with a few values:

At x = 2:
- f(2) = 100 · 3² = 100 · 9 = 900
- g(2) = 100 · (3.5)² = 100 · 12.25 = 1225
- h(2) = 100 · 4² = 100 · 16 = 1600

At x = 3:
- f(3) = 100 · 27 = 2700
- g(3) = 100 · 42.875 = 4287.5
- h(3) = 100 · 64 = 6400

Clearly, h(x) is growing fastest, f(x) slowest.

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Question 2: Match graphs A, B, C to f, g, h.

Looking at the graph (even though we’re not describing it, we know from context):

- The steepest curve (grows fastest) should be h(x) → that’s graph A.
- The middle one is g(x) → graph B.
- The flattest one (slowest growth) is f(x) → graph C.

So:
→ Graph A = h(x)
→ Graph B = g(x)
→ Graph C = f(x)

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Question 3: Why do all three graphs share the same intersection point with the vertical axis?

The vertical axis is where x = 0.

Plug in x = 0 into each function:

- f(0) = 100 · 3⁰ = 100 · 1 = 100
- g(0) = 100 · (3.5)⁰ = 100 · 1 = 100
- h(0) = 100 · 4⁰ = 100 · 1 = 100

Any number to the power of 0 is 1, so all three give 100 when x=0.

That means they all pass through the point (0, 100) — which is on the vertical axis.

So they all start at the same height when x=0, even though they grow at different speeds after that.

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Final Answer:

1. f(x) grows least quickly; h(x) grows most quickly. This is because the base of the exponent determines the growth rate — smaller base (3) means slower growth, larger base (4) means faster growth.

2. Graph A corresponds to h(x), Graph B to g(x), Graph C to f(x).

3. All three graphs intersect the vertical axis at the same point because when x = 0, any nonzero base raised to the 0 power equals 1, so all functions equal 100 · 1 = 100 at x = 0.