Solutions — Graphs of Exponential Functions

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1. Q1 — Growth or decay, y-intercept

   (a) y=3^x: base 3>1, growth; y-intercept (0,1).   (b) y=(0.5)^x: base 0.5<1, decay; y-intercept (0,1).   (c) y=(3/4)^x: base 3/4<1, decay; y-intercept (0,1).   (d) y=5^x: base 5>1, growth; y-intercept (0,1).   (e) y=(1.2)^x: base 1.2>1, growth; y-intercept (0,1).

2. Q2 — Tables of values

   (a) y=2^x: x=−2→0.25; x=−1→0.5; x=0→1; x=1→2; x=2→4; x=3→8. Increasing curve through (0,1).   (b) y=3^x: x=−2→1/9; x=−1→1/3; x=0→1; x=1→3; x=2→9. Steeper than 2^x.   (c) y=(1/3)^x: x=−2→9; x=−1→3; x=0→1; x=1→1/3; x=2→1/9. Decreasing curve through (0,1).

3. Q3 — Asymptote, y-intercept, domain and range

   (a) y=2^x+3: asymptote y=3; y-intercept (0,4); domain all reals; range y>3.   (b) y=3^x−5: asymptote y=−5; y-intercept (0,−4); domain all reals; range y>−5.   (c) y=−2^x: asymptote y=0; y-intercept (0,−1); domain all reals; range y<0.   (d) y=4×2^x: asymptote y=0; y-intercept (0,4); domain all reals; range y>0.

4. Q4 — x-intercepts

   (a) 2^x−4=0 → 2^x=4=2² → x=2. x-intercept: (2,0).   (b) 3×2^x=12 → 2^x=4 → x=2. x-intercept: (2,0).   (c) 2^x=1/4=2⁻² → x=−2. x-intercept: (−2,0).   (d) −2^x+8=0 → 2^x=8=2³ → x=3. x-intercept: (3,0).

5. Q5 — Describing transformations from y=2^x

   (a) y=2^x−3: right 3. Asymptote y=0; y-intercept: y=2⁻³=1/8, so (0,1/8).   (b) y=2×2^x+1: vertical dilation factor 2, shift up 1. Asymptote y=1; y-intercept: y=2+1=3, so (0,3).   (c) y=−2^x+1+4: left 1, reflect in x-axis, shift up 4. Asymptote y=4; y-intercept: y=−2¹+4=2, so (0,2).   (d) y=2^−x: reflect in y-axis. Same as y=(1/2)^x. Asymptote y=0; y-intercept (0,1).

6. Q6 — Matching

   (a) Increasing, y-int (0,1), asymptote y=0: A) y=2^x.   (b) Decreasing, y-int (0,1), asymptote y=0: B) y=(1/2)^x.   (c) Increasing, y-int (0,3), asymptote y=2: C) y=2^x+2.   (d) Decreasing, y-int (0,−1), asymptote y=0: D) y=−2^x.

7. Q7 — Finding the equation from a graph

   (a) y=b×2^x. At (0,5): b=5. At (1,10): y=5×2=10 ✓. Equation: y=5×2^x.   (b) y=b×3^x+2. At (0,5): b+2=5 → b=3. Check (1,11): 3×3+2=11 ✓. Equation: y=3×3^x+2.   (c) y=A×(1/2)^x. At (2,3): A×(1/4)=3 → A=12.

8. Q8 — Comparing rates of growth

   (a) Table: x=0: 2⁰=1, 3⁰=1; x=1: 2, 3; x=2: 4, 9; x=3: 8, 27; x=5: 32, 243. y=3^x grows faster for large positive x.   (b) 3^x−2^x>100. At x=5: 243−32=211>100. At x=4: 81−16=65<100. So first at x=5.   (c) Both pass through (0,1); y=3^x is steeper for x>0; y=2^x decreases more slowly for x<0.

9. Q9 — Bacteria population

   (a) At t=0: P=500×2⁰=500 bacteria.   (b) At t=4: P=500×2⁴=500×16=8000 bacteria.   (c) Asymptote is P=0 as t→−∞. In context, this means the bacteria count approaches zero for very early (past) times, but in practice the model is only valid from t≥0. As t→+∞ there is no upper asymptote — the population grows without bound.   (d) Key points: (0,500) and (4,8000). Graph is increasing and curves upward steeply.

10. Q10 — Investigating transformations

    (a) y=2^x+k. At x=0: 2^k=8=2³ → k=3. Two forms: y=2^x+3 and y=8×2^x.   (b) y=2^x+2=2^x×2²=4×2^x ✓. They are identical.   (c) At (0,3): a=3. At (2,75): 3×b²=75 → b²=25 → b=5 (b>0). a=3, b=5. Since b=5>1: exponential growth.