Graphing Exponential Functions.ks-ia2 - Kuta Software - Free Printable

Explanation:
We are asked to sketch the graphs of four exponential functions. Since this is a written response (not a drawing), I’ll describe how to plot each one step by step, including key points and behavior—so you can accurately draw them on the grids.

Recall: An exponential function has the form
 y = a · bˣ
where:
- a is the vertical stretch/compression and sign (if negative, reflection over x-axis),
- b > 0, b ≠ 1 is the base,
- if b > 1 → exponential growth (increases as x increases),
- if 0 < b < 1 → exponential decay (decreases as x increases).

Let’s handle each function:

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1) y = 4 · 2ˣ

Base = 2 (>1) → growth
a = 4 → starts at y = 4 when x = 0.

Compute key points:
- x = −2: y = 4 · 2⁻² = 4 · (1/4) = 1
- x = −1: y = 4 · 2⁻¹ = 4 · (1/2) = 2
- x = 0: y = 4 · 2⁰ = 4 · 1 = 4
- x = 1: y = 4 · 2¹ = 8
- x = 2: y = 4 · 2² = 16

So points: (−2, 1), (−1, 2), (0, 4), (1, 8), (2, 16)
Graph rises steeply to the right, approaches 0 as x → −∞ (horizontal asymptote y = 0).

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2) y = 5 · 2ˣ

Same base (growth), but a = 5.

Points:
- x = −2: y = 5 · 1/4 = 1.25
- x = −1: y = 5 · 1/2 = 2.5
- x = 0: y = 5
- x = 1: y = 10
- x = 2: y = 20

Points: (−2, 1.25), (−1, 2.5), (0, 5), (1, 10), (2, 20)
Same shape as #1, but vertically stretched (higher at every x).

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3) y = 4 · (½)ˣ

Note: (½)ˣ = 2⁻ˣ, so this is decay (since base < 1).
Alternatively, rewrite: y = 4 · 2⁻ˣ — same as #1 reflected over y-axis.

Points:
- x = −2: y = 4 · (½)⁻² = 4 · 4 = 16
- x = −1: y = 4 · (½)⁻¹ = 4 · 2 = 8
- x = 0: y = 4 · 1 = 4
- x = 1: y = 4 · ½ = 2
- x = 2: y = 4 · ¼ = 1

Points: (−2, 16), (−1, 8), (0, 4), (1, 2), (2, 1)
Graph falls to the right, rises to the left; still asymptotic to y = 0 as x → +∞.

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4) y = 2 · (½)ˣ = 2 · 2⁻ˣ

Decay, but smaller vertical stretch (a = 2).

Points:
- x = −2: y = 2 · (½)⁻² = 2 · 4 = 8
- x = −1: y = 2 · 2 = 4
- x = 0: y = 2
- x = 1: y = 2 · ½ = 1
- x = 2: y = 2 · ¼ = 0.5

Points: (−2, 8), (−1, 4), (0, 2), (1, 1), (2, 0.5)
Same shape as #3, but lower (scaled down by factor 2/4 = 0.5).

To sketch:
- Plot the 5 points for each.
- Draw a smooth curve through them, approaching y = 0 but never touching it.
- For growth functions (#1, #2), curve goes up fast to the right.
- For decay functions (#3, #4), curve goes down fast to the right.

Since the question only asks to *sketch*, and no specific answer format is required beyond that, the final answer is the set of correct graphs — but since we’re in text, the best we can do is confirm the key points above are correct.

Let me double-check calculations:

1) y = 4·2ˣ:
x=−2 → 4·(1/4)=1 ✓
x=0 → 4 ✓
x=2 → 4·4=16 ✓

2) y = 5·2ˣ:
x=0 → 5 ✓
x=1 → 10 ✓
x=−1 → 5·0.5=2.5 ✓

3) y = 4·(1/2)ˣ:
x=1 → 4·0.5=2 ✓
x=2 → 4·0.25=1 ✓
x=−1 → 4·2=8 ✓

4) y = 2·(1/2)ˣ:
x=0 → 2 ✓
x=2 → 2·0.25=0.5 ✓
All correct.

Final Answer:
The graphs are exponential curves with the following key points:
1) (−2,1), (−1,2), (0,4), (1,8), (2,16)
2) (−2,1.25), (−1,2.5), (0,5), (1,10), (2,20)
3) (−2,16), (−1,8), (0,4), (1,2), (2,1)
4) (−2,8), (−1,4), (0,2), (1,1), (2,0.5)
All approach y = 0 as x → ±∞ depending on growth/decay direction.