Roots, intercepts and turning points of quadratics worksheet — GCSE Higher
Fifteen questions on "roots, intercepts and turning points of quadratics" — DfE statement A11. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
Roots, intercepts and turning points of quadratics worksheet — GCSE Higher
Maths · GCSE Higher · A11 · Roots, intercepts and turning points of quadratics · 15 questions
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1.By completing the square, find the turning point of the curve y = x² + 6x + 2.
y = x² + 6x + 2
(a)x = 3, y = −7
(b)x = −3, y = −7
(c)x = −3, y = 7
(d)x = −6, y = −34
2.A quadratic graph y = ax² + bx + c has its turning point on the y-axis. Which statement about its roots must be true?
(a)The two roots must both be positive.
(b)The graph has no roots at all.
(c)The graph must touch the x-axis at exactly one point.
(d)If the graph has two real roots, they are equal and opposite in value, so they sum to zero.
3.By completing the square, find the turning point of the curve y = 3x² + 12x + 7.
y = 3x² + 12x + 7
(a)x = −2, y = 5
(b)x = −4, y = −41
(c)x = 2, y = −5
(d)x = −2, y = −5
4.The graph of y = x² − 7x + 2 crosses the x-axis at two points. One root, read from the graph, is approximately x = 0.30. Using the fact that the sum of the two roots of x² − 7x + 2 = 0 is 7, estimate the other root, correct to 2 decimal places.
y = x² − 7x + 2
(a)x ≈ 6.70
(b)x ≈ 6.30
(c)x ≈ 7.30
(d)x ≈ 0.70
5.A curve has equation y = x² − 9. Which statement about its graph is correct?
y = x² − 9
(a)It crosses the x-axis only once, at x = 9.
(b)It does not cross the x-axis, since −9 is negative.
(c)It crosses the x-axis at x = 9 and x = −9.
(d)It crosses the x-axis at x = 3 and x = −3.
6.The curve y = 2x² − 12x + 7 has a minimum point. By completing the square, find the value of x at which the minimum occurs.
y = 2x² − 12x + 7
(a)6
(b)−3
(c)−12
(d)3
7.Write y = x² + 12x + 40 in the form (x + a)² + b, and hence write down the minimum value of y.
y = x² + 12x + 40
(a)−6
(b)36
(c)4
(d)40
8.By completing the square, find the turning point of the curve y = x² − 4x + 9.
y = x² − 4x + 9
(a)x = 4, y = −7
(b)x = 2, y = −5
(c)x = 2, y = 5
(d)x = −2, y = 5
9.The curve y = −x² + 6x − 5 has a maximum point. Use completing the square to find its coordinates.
y = −x² + 6x − 5
(a)x = 3, y = −4
(b)x = 3, y = 4
(c)x = 6, y = 31
(d)x = −3, y = 4
10.By completing the square, find the turning point of the curve y = 2x² − 8x + 3.
y = 2x² − 8x + 3
(a)x = 4, y = −29
(b)x = 2, y = −5
(c)x = −2, y = −5
(d)x = 2, y = 5
11.Which of these quadratic graphs does NOT cross the x-axis at all?
(a)y = (x − 2)² + 3
(b)y = (x + 4)(x − 1)
(c)y = (x − 2)(x + 3)
(d)y = x² − 9
12.By completing the square, find the turning point of the curve y = x² + 8x − 3.
y = x² + 8x − 3
(a)x = −8, y = −67
(b)x = −4, y = 19
(c)x = 4, y = −19
(d)x = −4, y = −19
13.A quadratic graph has equation y = (x − 4)². A student says this graph crosses the x-axis at two different points. Explain why the student is wrong.
(a)It touches the x-axis once, only at x = 4.
(b)It never touches the x-axis at all.
(c)It crosses twice, at x = 4 and x = −4.
(d)It crosses twice, at x = 2 and x = −2.
14.A graph has equation y = x² − 6x + 5. A student says its turning point has x-coordinate 6, because that's the coefficient of x. Which statement corrects the student's mistake?
y = x² − 6x + 5
(a)The student is correct, because the turning point's x-coordinate always equals the coefficient of x.
(b)The turning point has x-coordinate −6, because the sign of the coefficient must be reversed.
(c)The turning point has x-coordinate 5, because that is the larger of the equation's two roots.
(d)The roots of x² − 6x + 5 = 0 are x = 1 and x = 5 (since it factorises to (x − 1)(x − 5)), so by symmetry the turning point has x-coordinate 3, not 6.
15.By completing the square, find the turning point of the curve y = x² − 10x + 30.