11.4 Quadratic Functions

**Factorising.** Solve each quadratic by factorising. (a) $x^2 - 7x + 12 = 0$ (b) $2x^2 - 5x - 3 = 0$ (c) $x^2 - 9 = 0$

**Completing the square.** Let $f(x) = x^2 + 8x + 10$. (a) Express $f(x)$ in vertex form. (b) Hence solve $f(x) = 0$, giving exact answers in surd form. (c) State the range of $f$. (d) Describe the single transformation that maps $y = x^2$ onto $y = (x + 5)^2$.

**Quadratic formula.** Solve, giving exact answers. (a) $x^2 - 4x + 1 = 0$ (b) $2x^2 + 5x - 3 = 0$ (c) $3x^2 - 7x + 2 = 0$

**Graph features.** For $f(x) = x^2 - 6x + 8$: (a) Find the $x$- and $y$-intercepts. (b) Find the vertex. (c) State the axis of symmetry. (d) Sketch the graph.

**Quadratic modelling — projectile.** A ball is thrown vertically. Its height (m) above ground after $t$ seconds is $h(t) = -5t^2 + 20t + 1.5$. (a) Find the height at $t = 0$ and explain what it represents. (b) Find the time at which the ball reaches its maximum height. (c) Find the maximum height. (d) Find, to 3 s.f., the time at which the ball hits the ground.

**Quadratic modelling — fountain.** A water jet follows $h(x) = -(x - 4)^2 + 9$ ($h$ height in m, $x$ horizontal distance in m). (a) Find the maximum height and where it occurs. (b) Find $h(0)$ and explain in context. (c) Find the $x$-values at which the jet returns to ground level.

**Optimisation — area.** A farmer uses 80 m of fencing to enclose a rectangular field with one side along a straight river (no fencing needed on that side). (a) Let the width perpendicular to the river be $x$ m. Express the length along the river and the area in terms of $x$. (b) Find the value of $x$ that maximises the area. (c) State the maximum area and the corresponding length along the river.

**Discriminant — tangency [EXT].** The line $y = 2x + 1$ meets the curve $y = x^2 + k$ at exactly one point. (a) Show that $x^2 - 2x + (k - 1) = 0$. (b) Find $k$ for which the line is tangent. (c) Find the coordinates of the point of tangency.

**Inequality [EXT].** Solve $x^2 - x - 6 \geq 0$. Express your answer in interval notation, and on a number line.

**Quadratic from data.** A quadratic $f(x) = ax^2 + bx + c$ has $f(0) = 5$, $f(1) = 6$, $f(2) = 13$. (a) Set up three equations in $a$, $b$, $c$. (b) Solve for $a$, $b$, $c$. (c) State the axis of symmetry and the vertex.

**Word problem — consecutive integers.** The product of two consecutive positive integers is 156. (a) Let the smaller be $n$. Write a quadratic equation in $n$. (b) Solve to find the integers.

**Two-variable optimisation.** A rectangular poster of total area 200 cm$^2$ has a 2 cm margin on all four sides. The printed area inside the margins is to be maximised, subject to the poster's overall dimensions being integer multiples of 1 cm. (a) Let the poster have width $w$ cm and height $\frac{200}{w}$ cm. Write the printed area $A$ in terms of $w$. (b) Sketch (or describe) $A(w)$ for $w > 4$ and find the value of $w$ that maximises $A$. (c) State the maximum printed area.