1
Review: powers, exponential equations, the quadratic function and zeros
Mathematics Prep 3 Term 1, Units 1–4, Review
GoalMix the ideas of Units 1 and 2: powers, exponential equations, the parabola and zeros.
New words
same base · write both sides with one base, then compare exponentsvertex · the turning point of a parabola, at x = −b/2azero of a function · a value of x for which f(x) = 0factorise · write an expression as a product
Explanation
Quick map of Units 1 and 2. Powers: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, and a⁻ⁿ = 1/aⁿ. Exponential equation: make the bases equal, then the exponents are equal: 3^(x+1) = 81 = 3⁴ gives x + 1 = 4, x = 3. Quadratic function f(x) = ax² + bx + c: vertex at x = −b/2a, axis of symmetry through it, y-intercept c, and minimum if a > 0 (maximum if a < 0). Zeros: solve f(x) = 0 by factorising: x² − 4x − 5 = (x − 5)(x + 1) so the zeros are 5 and −1. The picture shows this parabola: vertex (2, −9), so the minimum value is −9.
Worked examples
Simplify (2³ × 2⁵) ÷ 2⁶ = 2^(3+5−6) = 2² = 4.
Solve 5^(2x−1) = 125. 125 = 5³, so 2x − 1 = 3 and x = 2. Check: 5³ = 125 ✓.
p(x) = x³ − 4x = x(x − 2)(x + 2), so the zeros are 0, 2 and −2.
Class activity
Parabola fast-check. Write any quadratic with zeros 1 and 3. Find its vertex by symmetry (halfway between the zeros) and its minimum value.
Practice
1
Solve 4^x = 1/16.
1/16 = 4⁻², so x = −2.
2
Find the vertex and the minimum of f(x) = x² − 6x + 5.
x = 3, f(3) = −4: vertex (3, −4), minimum −4.
3
Find the zeros of x² − x − 6.
(x − 3)(x + 2) = 0: 3 and −2.
4
x = 5 is a zero of x² + kx − 10. Find k.
25 + 5k − 10 = 0, so k = −3.
5
Solve 2^t = 32.
32 = 2⁵, so t = 5.