Sketching Curves — Solutions

Full worked solutions, with method and final answer for every part.

Question 1

(a)y = x² has its minimum where x = 0.

(0, 0)

Question 2

(a)In y = (x−a)²+b form, the vertex is at (a, b). Here a=2, b=1.

Vertex (2, 1); the graph is y = x² translated 2 units right and 1 unit up

Question 3

(a)In y = −(x−a)²+b form, vertex is (a,b); the negative sign means the parabola opens downward, giving a maximum.

Vertex (1, 4), a maximum point

Question 4

(a)Adding 3 shifts every point on y=x² upward by 3 units.

y = x² + 3 is y = x² translated 3 units up

Question 5

(a)The curve crosses the x-axis where each bracket is zero.

x = −2 and x = 4, so (−2, 0) and (4, 0)

Question 6

(a)The line of symmetry of x²+bx+c passes through x = −b/2.

x = 3 (using −(−6)/2 = 3)

Question 7

(a)In (x−a)² form, vertex is (a, 0); here a=−3.

(−3, 0)

Question 8

(a)The coefficient 2 makes the curve rise more steeply than y=x² for the same x.

y = 2x² is narrower/steeper than y = x²

Question 9

(a)The line of symmetry passes through the vertex's x-coordinate.

x = 4

Question 10

(a)The negative sign in front of x² means the parabola opens downward.

Maximum point at (0, 5)

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