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Problem-solving Pack

Ecolint Campus des NationsMathematics
Year 11 · 11.4 Quadratic Functions

Problem-solving Pack

Name: _________________________________
Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 12
Factorising. Solve each quadratic by factorising.

(a) x2−7x+12=0x^2 - 7x + 12 = 0
(b) 2x2−5x−3=02x^2 - 5x - 3 = 0
(c) x2−9=0x^2 - 9 = 0

Working space

2Problem 2 of 12
Completing the square. Let f(x)=x2+8x+10f(x) = x^2 + 8x + 10.

(a) Express f(x)f(x) in vertex form.
(b) Hence solve f(x)=0f(x) = 0, giving exact answers in surd form.
(c) State the range of ff.
(d) Describe the single transformation that maps y=x2y = x^2 onto y=(x+5)2y = (x + 5)^2.

Working space

3Problem 3 of 12
Quadratic formula. Solve, giving exact answers.

(a) x2−4x+1=0x^2 - 4x + 1 = 0
(b) 2x2+5x−3=02x^2 + 5x - 3 = 0
(c) 3x2−7x+2=03x^2 - 7x + 2 = 0

Working space

4Problem 4 of 12
Graph features. For f(x)=x2−6x+8f(x) = x^2 - 6x + 8:

(a) Find the xx- and yy-intercepts.
(b) Find the vertex.
(c) State the axis of symmetry.
(d) Sketch the graph.

Working space

5Problem 5 of 12
Quadratic modelling — projectile. A ball is thrown vertically. Its height (m) above ground after tt seconds is h(t)=−5t2+20t+1.5h(t) = -5t^2 + 20t + 1.5.

(a) Find the height at t=0t = 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.

Working space

6Problem 6 of 12
Quadratic modelling — fountain. A water jet follows h(x)=−(x−4)2+9h(x) = -(x - 4)^2 + 9 (hh height in m, xx horizontal distance in m).

(a) Find the maximum height and where it occurs.
(b) Find h(0)h(0) and explain in context.
(c) Find the xx-values at which the jet returns to ground level.

Working space

7Problem 7 of 12
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 xx m. Express the length along the river and the area in terms of xx.
(b) Find the value of xx that maximises the area.
(c) State the maximum area and the corresponding length along the river.

Working space

8Problem 8 of 12
Discriminant — tangency [EXT]. The line y=2x+1y = 2x + 1 meets the curve y=x2+ky = x^2 + k at exactly one point.

(a) Show that x2−2x+(k−1)=0x^2 - 2x + (k - 1) = 0.
(b) Find kk for which the line is tangent.
(c) Find the coordinates of the point of tangency.

Working space

9Problem 9 of 12
Inequality [EXT]. Solve x2−x−6≥0x^2 - x - 6 \geq 0. Express your answer in interval notation, and on a number line.

Working space

10Problem 10 of 12
Quadratic from data. A quadratic f(x)=ax2+bx+cf(x) = ax^2 + bx + c has f(0)=5f(0) = 5, f(1)=6f(1) = 6, f(2)=13f(2) = 13.

(a) Set up three equations in aa, bb, cc.
(b) Solve for aa, bb, cc.
(c) State the axis of symmetry and the vertex.

Working space

11Problem 11 of 12
Word problem — consecutive integers. The product of two consecutive positive integers is 156.

(a) Let the smaller be nn. Write a quadratic equation in nn.
(b) Solve to find the integers.

Working space

12Problem 12 of 12
Two-variable optimisation. A rectangular poster of total area 200 cm2^2 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 ww cm and height 200w\frac{200}{w} cm. Write the printed area AA in terms of ww.
(b) Sketch (or describe) A(w)A(w) for w>4w > 4 and find the value of ww that maximises AA.
(c) State the maximum printed area.

Working space