Corrigé
11.4 Quadratic Functions
Pack A — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Solve $x^2 - 7x + 12 = 0$ by factorising. | $x = 3$ or $x = 4$ |
| 2 | Solve $x^2 - 16 = 0$. | $x = \pm 4$ |
| 3 | Solve $x^2 + 5x + 6 = 0$ using factorising. | $x = -2$ or $x = -3$ |
| 4 | State the coordinates of the vertex of $y = (x - 3)^2 + -4$ and whether it is a min or max. | Vertex $(3, -4)$; minimum |
| 5 | State the $y$-intercept of $y = x^2 - 5x + 7$. | $(0, 7)$ |
| 6 | Solve $2x^2 - 5x - 3 = 0$ by factorising. | $x = -\dfrac{1}{2}$ or $x = 3$ |
| 7 | State the axis of symmetry of $y = x^2 - 6x + 8$. | $x = 3$ |
| 8 | Solve $x^2 - 4x - 5 = 0$ using the quadratic formula. | $x = -1$ or $x = 5$ |
| 9 | Does $y = -2x^2 + 3x + 1$ open upwards or downwards? | Downwards (because the leading coefficient is negative). |
| 10 | State the $x$-intercepts of $y = (x - 2)(x + 5)$. | $x = 2$ and $x = -5$ |
| 11 | Express $x^2 + 8x + 10$ in the form $(x + p)^2 + q$. | $(x + 4)^2 - 6$ |
| 12 | Find the vertex of $y = x^2 - 6x + 11$. | $(3, 2)$ |
| 13 | Solve $2x^2 - x - 6 = 0$ by factorising. | $x = -\dfrac{3}{2}$ or $x = 2$ |
| 14 | Expand $(x - 3)(x + 5)$ and write in $ax^2 + bx + c$ form. | $x^2 + 2x - 15$ |
| 15 | Write $y = 2(x - 3)^2 - 5$ in expanded form $y = ax^2 + bx + c$. | $y = 2x^2 - 12x + 13$ |
| 16 | Solve $x^2 + 8x + 10 = 0$ exactly, leaving any irrational answer in surd form. | $x = -4 \pm \sqrt{6}$ |
| 17 | A number plus its square is 30. Find the number. | $x = 5$ or $x = -6$ |
| 18 | For $y = (x - 3)^2 - 4$, state (a) the vertex, (b) the $x$-intercepts. | (a) $(3, -4)$; (b) $x = 1$ and $x = 5$ |
| 19 | Express $y = 2x^2 - 8x + 5$ in vertex form. | $y = 2(x - 2)^2 - 3$ |
| 20 | A rectangle has length $(x + 3)$ cm and width $x$ cm. Its area is 18 cm$^2$. Find $x$. | $x = 3$ cm |
| 21 | Express $f(x) = x^2 + 8x + 10$ in vertex form, then state the range of $f$. | $(x + 4)^2 - 6$; range $y \geq -6$ |
| 22 | A quadratic $y = x^2 + bx + 7$ has its minimum value at $x = 3$. Find $b$. | $b = -6$ |
| 23 | How many real roots does $2x^2 - 4x + 3 = 0$ have? Use the discriminant. | No real roots ($\Delta = 16 - 24 = -8 < 0$). |
| 24 | The line $y = 2x + 1$ is tangent to the curve $y = x^2 + k$. Find $k$. | $k = 2$ |
| 25 | Solve $x^2 - 5x + 6 \leq 0$. | $2 \leq x \leq 3$ |
| 26 | A monic quadratic has roots $-2$ and $5$. Write it in expanded form. | $x^2 - 3x - 10$ |
| 27 | A ball thrown follows $h(t) = -5t^2 + 20t + 1$ ($h$ in m, $t$ in s). Find (a) the maximum height, (b) the time it occurs. | (a) 21 m; (b) at $t = 2$ s |
| 28 | For $x^2 + 5x - 6 = 0$, find the sum and product of the roots without solving. | Sum $-5$; product $-6$ |
| 29 | A fountain jet has $h(x) = -(x - 4)^2 + 9$ ($h$ height in m, $x$ horizontal distance in m). Find (a) max height, (b) range of $x$ above ground. | (a) 9 m; (b) $1 \leq x \leq 7$ |
| 30 | For what range of $k$ does $x^2 + kx + 9 = 0$ have no real roots? | $-6 < k < 6$ |
| 31 | A rectangle has perimeter 26 cm and area 40 cm$^2$. Find its dimensions. | 5 cm by 8 cm |
| 32 | For what values of $k$ does the line $y = x + k$ intersect $y = x^2$ at two distinct points? | $k > -\dfrac{1}{4}$ |
| 33 | A quadratic $y = ax^2 + bx + c$ passes through $(0, 5)$, $(1, 8)$ and $(3, 2)$. Find $a$, $b$, $c$. | $a = -\dfrac{5}{2}$, $b = \dfrac{11}{2}$, $c = 5$ |
| 34 | Express $y = x^2 - 6x + 11$ in vertex form. Hence describe the transformation from $y = x^2$ to $y = x^2 - 6x + 11$. | $y = (x - 3)^2 + 2$; translation 3 right and 2 up. |
| 35 | A parabola has vertex $(2, 3)$ and passes through $(5, 30)$. Find its equation. | $y = 3(x - 2)^2 + 3$ |
| 36 | Find the points of intersection of $y = x^2 + 2x - 3$ and $y = x + 1$. | $(-4, -3)$ and $(1, 2)$ |
| 37 | Find the minimum value of $f(x) = 3x^2 - 12x + 7$. | $-5$ |
| 38 | Show that for all real $k$, the equation $x^2 + 2kx + (k^2 + 1) = 0$ has no real solutions. | $\Delta = (2k)^2 - 4(k^2 + 1) = -4 < 0$. |
| 39 | A farmer has 40 m of fencing to enclose a rectangular pen against a wall (so one side is the wall). Find the dimensions giving the maximum area. | 20 m parallel to wall, 10 m perpendicular; area 200 m$^2$ |
| 40 | A ball is thrown and its height (m) at times $t = 0, 1, 2$ is recorded as $1.5, 5, 7.5$ m. Assuming $h(t) = at^2 + bt + c$, find $a$, $b$, $c$ and predict $h(3)$. | $a = -0.5$, $b = 4$, $c = 1.5$; $h(3) = 9$ |
Pack B — Réponses
| # | Question | Réponse |
|---|---|---|
| 1 | Solve $x^2 - 9x + 20 = 0$ by factorising. | $x = 4$ or $x = 5$ |
| 2 | Solve $x^2 - 25 = 0$. | $x = \pm 5$ |
| 3 | Solve $x^2 + 5x + 6 = 0$ using factorising. | $x = -2$ or $x = -5$ |
| 4 | State the coordinates of the vertex of $y = (x - -2)^2 + 5$ and whether it is a min or max. | Vertex $(-2, 5)$; minimum |
| 5 | State the $y$-intercept of $y = x^2 - -3x + 4$. | $(0, 4)$ |
| 6 | Solve $2x^2 - 5x - 3 = 0$ by factorising. | $x = -1$ or $x = \dfrac{2}{3}$ |
| 7 | State the axis of symmetry of $y = x^2 - 4x + -1$. | $x = 2$ |
| 8 | Solve $x^2 - 4x - 5 = 0$ using the quadratic formula. | $x = -4$ or $x = 2$ |
| 9 | Does $y = -2x^2 + 3x + 1$ open upwards or downwards? | Upwards. |
| 10 | State the $x$-intercepts of $y = (x - 4)(x + 3)$. | $x = 4$ and $x = -3$ |
| 11 | Express $x^2 + 6x + 5$ in the form $(x + p)^2 + q$. | $(x + 3)^2 - 4$ |
| 12 | Find the vertex of $y = x^2 - 6x + 11$. | $(-2, -5)$ |
| 13 | Solve $2x^2 - x - 6 = 0$ by factorising. | $x = -2$ or $x = \dfrac{1}{3}$ |
| 14 | Expand $(x - 3)(x + 5)$ and write in $ax^2 + bx + c$ form. | $2x^2 + 7x - 4$ |
| 15 | Write $y = 2(x - 3)^2 - 5$ in expanded form $y = ax^2 + bx + c$. | $y = -3x^2 - 6x + 1$ |
| 16 | Solve $x^2 + 8x + 10 = 0$ exactly, leaving any irrational answer in surd form. | $x = 3 \pm \sqrt{7}$ |
| 17 | A number plus its square is 30. Find the number. | $x = 7$ or $x = -8$ |
| 18 | For $y = (x - 3)^2 - 4$, state (a) the vertex, (b) the $x$-intercepts. | (a) $(-1, -9)$; (b) $x = -4$ and $x = 2$ |
| 19 | Express $y = 2x^2 - 8x + 5$ in vertex form. | $y = 3(x + 2)^2 - 13$ |
| 20 | A rectangle has length $(x + 3)$ cm and width $x$ cm. Its area is 18 cm$^2$. Find $x$. | $x = 3$ cm |
| 21 | Express $f(x) = x^2 + 8x + 10$ in vertex form, then state the range of $f$. | $(x - 3)^2 - 7$; range $y \geq -7$ |
| 22 | A quadratic $y = x^2 + bx + 7$ has its minimum value at $x = 3$. Find $b$. | $b = 4$ |
| 23 | How many real roots does $2x^2 - 4x + 3 = 0$ have? Use the discriminant. | One repeated real root ($\Delta = 36 - 36 = 0$). |
| 24 | The line $y = 2x + 1$ is tangent to the curve $y = x^2 + k$. Find $k$. | $k = 1$ |
| 25 | Solve $x^2 - 5x + 6 \leq 0$. | $x < -3$ or $x > 4$ |
| 26 | A monic quadratic has roots $-2$ and $5$. Write it in expanded form. | $x^2 + x - 12$ |
| 27 | A ball thrown follows $h(t) = -5t^2 + 20t + 1$ ($h$ in m, $t$ in s). Find (a) the maximum height, (b) the time it occurs. | (a) 47 m; (b) at $t = 3$ s |
| 28 | For $x^2 + 5x - 6 = 0$, find the sum and product of the roots without solving. | Sum $\dfrac{3}{2}$; product $-2$ |
| 29 | A fountain jet has $h(x) = -(x - 4)^2 + 9$ ($h$ height in m, $x$ horizontal distance in m). Find (a) max height, (b) range of $x$ above ground. | (a) 16 m; (b) $-1 \leq x \leq 7$ |
| 30 | For what range of $k$ does $x^2 + kx + 9 = 0$ have no real roots? | $-4 < k < 4$ |
| 31 | A rectangle has perimeter 26 cm and area 40 cm$^2$. Find its dimensions. | 3 cm by 8 cm |
| 32 | For what values of $k$ does the line $y = x + k$ intersect $y = x^2$ at two distinct points? | $k > 0$ |
| 33 | A quadratic $y = ax^2 + bx + c$ passes through $(0, 5)$, $(1, 8)$ and $(3, 2)$. Find $a$, $b$, $c$. | $a = 3$, $b = 0$, $c = -2$ |
| 34 | Express $y = x^2 - 6x + 11$ in vertex form. Hence describe the transformation from $y = x^2$ to $y = x^2 - 6x + 11$. | $y = (x + 2)^2 - 5$; translation 2 left and 5 down. |
| 35 | A parabola has vertex $(2, 3)$ and passes through $(5, 30)$. Find its equation. | $y = 3(x + 1)^2 - 4$ |
| 36 | Find the points of intersection of $y = x^2 + 2x - 3$ and $y = x + 1$. | $(0, -1)$ and $(3, 2)$ |
| 37 | Find the minimum value of $f(x) = 3x^2 - 12x + 7$. | $-11$ |
| 38 | Show that for all real $k$, the equation $x^2 + 2kx + (k^2 + 1) = 0$ has no real solutions. | $\Delta = (-2k)^2 - 4(k^2 + 4) = -16 < 0$. |
| 39 | A farmer has 40 m of fencing to enclose a rectangular pen against a wall (so one side is the wall). Find the dimensions giving the maximum area. | 30 m parallel, 15 m perpendicular; area 450 m$^2$ |
| 40 | A ball is thrown and its height (m) at times $t = 0, 1, 2$ is recorded as $1.5, 5, 7.5$ m. Assuming $h(t) = at^2 + bt + c$, find $a$, $b$, $c$ and predict $h(3)$. | $a = -1$, $b = 5$, $c = 2$; $h(3) = 8$ |
Problèmes — Solutions détaillées
**Factorising.** Solve each quadratic by factorising. (a) $x^2 - 7x + 12 = 0$ (b) $2x^2 - 5x - 3 = 0$ (c) $x^2 - 9 = 0$
(a) $x = 3$ or $x = 4$. (b) $x = -\frac{1}{2}$ or $x = 3$. (c) $x = \pm 3$.
**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 $y = (x + 5)^2$.
(a) $(x + 4)^2 - 6$. (b) $x = -4 \pm \sqrt{6}$. (c) $y \geq -6$. (d) Translation 5 units left.
**Quadratic formula.** Solve, giving exact answers. (a) $x^2 - 4x + 1 = 0$ (b) $2x^2 + 5x - 3 = 0$ (c) $3x^2 - 7x + 2 = 0$
(a) $x = 2 \pm \sqrt{3}$. (b) $x = -3$ or $x = \frac{1}{2}$. (c) $x = 2$ or $x = \frac{1}{3}$.
**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.
(a) $x$-intercepts $(2, 0)$ and $(4, 0)$; $y$-intercept $(0, 8)$. (b) $(3, -1)$. (c) $x = 3$.
**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.
(a) 1.5 m — initial height (release point). (b) $t = 2$ s. (c) 21.5 m. (d) $t \approx 4.07$ s.
**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.
(a) 9 m at $x = 4$ m. (b) $h(0) = -7$ — nozzle is 7 m below ground. (c) $x = 1$ or $x = 7$.
**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.
(a) Length $= 80 - 2x$; area $A = x(80 - 2x) = 80x - 2x^2$. (b) $x = 20$. (c) Max area 800 m$^2$ with length 40 m.
**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.
(a) See working. (b) $k = 2$. (c) $(1, 3)$.
**Inequality [EXT].** Solve $x^2 - x - 6 \geq 0$. Express your answer in interval notation, and on a number line.
$x \leq -2$ or $x \geq 3$; i.e. $(-\infty, -2] \cup [3, \infty)$.
**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.
(a) $c = 5$; $a + b + c = 6$; $4a + 2b + c = 13$. (b) $a = 3$, $b = -2$, $c = 5$. (c) Axis $x = \frac{1}{3}$; vertex $\left(\frac{1}{3}, \frac{14}{3}\right)$.
**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.
(a) $n(n + 1) = 156$. (b) 12 and 13.
**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.
(a) $A = (w - 4)\left(\dfrac{200}{w} - 4\right) = 200 - 4w - \dfrac{800}{w} + 16 = 216 - 4w - \dfrac{800}{w}$. (b) Differentiating gives $w = \sqrt{200} \approx 14.14$, so $w = 14$ cm. (c) Approximately $130.3$ cm$^2$.