Parabolas
1. For each parabola below find
(i)
(ii)
(iii)
the point of crossing the y-axis
the roots of the parabola
the minimum or maximum turning point
y = x2 – 6x
(a)
y = x2 + 4x
(b)
y
y
x
x
(c)
(d)
y
y=8
y = x2 – 9
y
x
x
y = 8x – x2
y = 2x2 – 32
(e)
y = (f)
8x - x2
y
y
x
x
y = 25 – x2
y = x2 – 6x + 8
(g)
y = x2 + 8x + 15
(h)
y
y
x
x
y = x2 + 2x - 15
(i)
y = x2 – 4x – 12
(j)
y
y
x
(k)
y
(l)
x
y = 2x2 + 5x – 3
y
x
y = -x2 + 10x - 9
x
y
2. The diagram shows the parabola
y = x2 + 2x.
(a) Find the coordinates of the point A.
(b) Find the coordinates of B, the minimum
turning point of the parabola.
x
A
B
y
B
3. The diagram shows the parabola
y = 12x – 2x2
(a)Find the coordinates of the point A.
(b)Find the coordinates of B, the maximum
turning point of the parabola.
x
A
y
4. The parabola with equation
y=4–x
C
2
is shown opposite.
x
A
(a) Find the coordinates of A and B,
the roots of the parabola.
(b) Find the coordinates of C.
B
y
5. The diagram shows the graph of
x
A
B
y = 3x – 27
2
(a) Find A and B.
(b) Find the coordinates of C,
the minimum turning point.
C
y
6. The diagram opposite shows part of the graph of
B
y = x2 – 8x – 9.
C
x
The graph cuts the y-axis at A and the x-axis at B
and C.
A
(a) Write down the coordinates of A
(b) Find the coordinates of B and C
(c) Calculate the minimum value of x2 – 8x – 9.
7. The diagram shows the parabola
E
y
y = x2 – 10x + 16
x
(a) Write down the coordinates of E
(b) Find the coordinates of F and G
(c) Find the coordinates of H, the minimum
turning point.
F
G
H
8. The parabola with equation
y
y = x2 – 4x – 5
is shown opposite.
(a) Write down the coordinates of A
(b) Find the coordinates of B and C
(c) Find the coordinates of D, the minimum
turning point.
x
C
B
A
D
y
9. The diagram shows the parabola
y = x2 – 10x – 11
(a) Write down the coordinates of A
(b) Find the coordinates of B and C
(c) Find the minimum value of
y = x2 – 10x – 11.
x
C
B
A
y
10. The graph of
y = x2 + 8x + 7
A
is shown opposite.
x
C
B
(a) Write down the coordinates of A
(b) Find the coordinates of B and C
(c) Find the coordinates of D, the minimum
turning point.
D
y
M
N
11. The diagram shows the parabola
y = – x2 – 2x + 15
x
(a) Write down the coordinates of N
(d) Find the coordinates of K and L
(e) Find the coordinates of M, the maximum
turning point.
L
K
y
12. The diagram shows the parabola
y = – x2 + 6x + 7
(a) Write down the coordinates of A
(b) Find the coordinates of B and C
(c) Find the maximum value of
y = – x2 + 6x + 7
A
x
B
C
y
13. The graph of
y = x2 – x – 2
is shown opposite.
x
C
B
A D
(a) Write down the coordinates of A
(b) Find the coordinates of B and C
(c) Find the coordinates of D, the minimum
turning point.
y
14. The graph of
y = x2 + 5x – 6
x
R
Q
is shown opposite.
T
(a) Write down the coordinates of T
(b) Find the coordinates of Q and R
(c) Find the coordinates of P, the minimum
turning point.
P
15. The diagram opposite shows part of the
graph of
y
y = 4x2 + 4x – 3.
The graph cuts the y-axis at A and the
x-axis at B and C.
x
B
B
(a) Write down the coordinates of A
(b) Find the coordinates of B and C.
(c) Calculate the minimum value of 4x2 + 4x – 3
D
C
A
y
16. The diagram opposite shows part of the graph of
y =  3x2 + 2x + 1.
P
The graph cuts the y-axis at P and the x-axis
at Q and R.
x
B
Q
R
(a) Write down the coordinates of P.
(b) Find the coordinates of Q and R.
(c) Find the maximum turning point of
the parabola.
17. The diagram opposite shows part of the
graph of y = k(x – a)(x – b).
The graph cuts the y-axis at (0,-6) and the
x-axis at (-1,0) and (3,0).
y
x
(a) Write down the values of a and b.
(b) Calculate the value of B
k.
(c) Find the coordinates of the minimum turning
point of the parabola.
-1
3
-6
D
y
D
18. The diagram opposite shows part of
the graph of y = k(x – a)(x – b).
The graph cuts the y-axis at (0,-18)
B
and the x-axis at (-3,0) and (2,0).
x
-3
2
(a) Write down the values of a and b.
(b) Calculate the value of k.
(c) Find the minimum value of the parabola.
-18
y
4
19. The diagram opposite shows part of
the graph of y = k(x + a)(x + b).
The graph cuts the y-axis at (0,4) and
B
the x-axis at (-1,0) and (2,0).
x
2
-1
(a) Write down the values of a and b.
(b) Find the value of k.
(c) Find the coordinates of the maximum
turning point of the parabola.
20. The diagram opposite shows part of
the graph of y = p(x + a)(x + b).
The graph cuts the y-axis at (0,-16)
and the x-axis at (-4,0) and (1,0).
(a) Write down the values of a and b.
(b) Find the value of p.
(c) Find the minimum value of y.
4
B
-4
y
x
-18
1
-16