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iGCSE (2021 Edition)

26.04 Equations of tangents (Extended)

Worksheet
Equation of a tangent from a graph
1

Each of the following graphs contain a curve, f \left( x \right), along with one of its tangents, g \left( x \right).

i

State the coordinates of the point at which g \left( x \right) is a tangent to the curve f \left( x \right).

ii

State the gradient of the tangent.

iii

Hence determine the equation of the line y = g \left( x \right).

a
-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
b
-3
-2
-1
1
2
3
x
-6
-5
-4
-3
-2
-1
1
2
3
y
c
1
2
3
4
5
x
-1
1
2
3
y
d
1
2
3
4
5
x
-3
-2
-1
1
2
y
e
-1
1
2
3
4
5
x
1
2
3
4
5
y
2

Each of the following graphs contain a curve, f \left( x \right), along with one of its tangents, g \left( x \right).

i

State the coordinates of the point at which g \left( x \right) is a tangent to the curve f \left( x \right).

ii

State the gradient of the tangent.

iii

Hence determine the equation of the line y = g \left( x \right).

iv

State the x-coordinate of the point on the curve at which we could draw a tangent that has the same gradient as g \left( x \right).

a
-2
-1
1
2
3
x
-6
-5
-4
-3
-2
-1
1
2
3
y
b
-3
-2
-1
1
2
x
-4
-3
-2
-1
1
2
3
4
5
y
3

Consider the graph of the function f \left( x \right):

a

Sketch the graph of the function \\ g \left( x \right) = 2 x + 3 on the same number plane.

b

Is g \left( x \right) a tangent to f \left( x \right)? Explain your answer.

-4
-3
-2
-1
1
2
3
4
x
1
2
3
4
5
6
7
8
9
y
4

Consider the graph of the function f \left( x \right):

a

Sketch the graph of the function \\ g \left( x \right) = 2 x-1 on the same number plane.

b

Is g \left( x \right) a tangent to f \left( x \right)? Explain your answer.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
y
5

Consider the graph of the function f \left( x \right):

a

Sketch the graph of the function \\ g \left( x \right) = 3 x + 3 on the same number plane.

b

Is g \left( x \right) a tangent to f \left( x \right)? Explain your answer.

-4
-3
-2
-1
1
2
3
4
x
-4
-3
-2
-1
1
2
3
4
5
y
6

Consider the graph of the function f \left( x \right):

a

Sketch the graph of the function \\ g \left( x \right) = 4 x + 7 on the same number plane.

b

Is g \left( x \right) a tangent to f \left( x \right)? Explain your answer.

-4
-3
-2
-1
1
2
x
-8
-6
-4
-2
2
4
6
8
y
7

Consider the curve given by the function f \left( x \right) = x^{2} - 1.

a

Find the gradient of the tangent to the curve at the point \left(1, 0\right).

b

Graph the curve and the tangent at the point \left(1, 0\right) on a number plane.

8

Consider the curve given by the function f \left( x \right) = x^{2} - 4 x + 2.

a

Find the gradient of the tangent to the curve at the point \left(3, -1\right).

b

State the coordinates of the vertex of the parabola f \left( x \right) = x^{2} - 4 x + 2.

c

Graph the curve and the tangent at the point \left(3, -1\right) on a number plane.

Equation of a tangent at a given point
9

Consider the function y = x^{2} - 3 x + 4. State the x-coordinate of the point on the curve where the tangent makes an angle of 45 \degree with the x-axis.

10

Consider the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 1 , 1\right).

a

Find the gradient of the function f \left( x \right) = x^{2} at this point.

b

Find the equation of the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 1 , 1\right).

11

Consider the tangent to the curve f \left( x \right) = x^{3} at the point \left( - 1 , -1\right).

a

Find the gradient of the function f \left( x \right) = x^{3} at this point.

b

Hence find the equation of the tangent to the curve f \left( x \right) = x^{3} at the point \left( - 1 , -1\right).

12

Consider the tangent to the curve f \left( x \right) = - x^{2} at the point \left( - 3 , - 9 \right).

a

Find the gradient of the function f \left( x \right) = - x^{2} at this point.

b

Hence find the equation of the tangent to the curve f \left( x \right) = - x^{2} at the point \left( - 3 , - 9 \right).

13

Consider the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 2 , 4\right).

a

Find the gradient of the function f \left( x \right) = x^{2} at this point.

b

Find the equation of the tangent to the curve f \left( x \right) = x^{2} at the point \left( - 2 , 4\right)

14

Consider the function f \left( x \right) = 3 x^{2}.

a

Find the gradient of the function at x = 2

b

Find the y-coordinate of the point on the function at x=2.

c

Hence find the equation of the tangent to the curve f \left( x \right) = 3 x^{2} at x = 2.

15

Find the equation of the tangent to the curve f \left( x \right) = 0.3 x^{3} - 5 x^{2} - x + 4 at x = 1.

Applications
16

For each of the following:

i

Find the x-coordinate of point M.

ii

Find the y-coordinate of point M.

a

At point M\left(x, y\right), the equation of the tangent to the curve y = x^{2} is given by y = 4 x - 4.

b

At point M\left(x, y\right), the equation of the tangent to the curve y = x^{3} is given by \\ y = 12 x - 16.

17

Consider the function f \left( x \right) = \dfrac{4 x^{3}}{3} + \dfrac{5 x^{2}}{2} - 3 x + 7. Find the x-coordinates of the points on the curve whose tangent is parallel to the line y = 3 x + 7.

18

5 x + y + 2 = 0 is the tangent to the curve y = x^{2} + b x + c at the point \left(9, - 47 \right).

a

Find the derivative \dfrac{d y}{d x} of y = x^{2} + b x + c.

b

Find the gradient of the tangent to the curve at x = 9.

c

Solve for the value of b.

d

Solve for the value of c.

19

Consider the function f \left( x \right) = x^{2} + 5 x.

a

Find the x-coordinate of the point at which f \left( x \right) has a gradient of 13.

b

Hence state the coordinates of the point on the curve where the gradient is 13.

20

Consider the function f \left( x \right) = x^{3} - 6 x^{2}.

a

Find the x-coordinates of the points at which f \left( x \right) has a gradient of 495.

b

Hence state the coordinates of the points on the curve where the gradient is 495.

21

Consider the function f \left( x \right) = x^{3} + 3 x^{2} - 19 x + 2.

a

Find the x-coordinates of the points at which f \left( x \right) has a gradient of 5.

b

Hence state the coordinates of the points on the curve where the gradient is 5.

22

Consider the function f \left( x \right) = x^{3} + 6 x^{2} - 14 x - 2.

a

Find the x-coordinates of the points on the curve where the gradient is the same as that of g \left( x \right) = 22 x - 4.

b

Hence state the coordinates of the points on the curve where the gradient is the same as that of g \left( x \right) = 22 x - 4.

23

Consider the function y = 4 x^{2} - 5 x + 2.

a

Find \dfrac{dy}{dx}.

b

Hence find the value of x at which the tangent to the parabola is parallel to the x-axis.

24

The curve y = a x^{3} + b x^{2} + 2 x - 17 has a gradient of 58 at the point \left(2, 31\right). Find the values of a and b.

25

The graph of y = a x^{3} + b x^{2} + c x + d intersects the x-axis at \left(2, 0\right), where it has a gradient of 36. It also intersects the y-axis at y = - 28, where the tangent is parallel to the x-axis.

Find the values of a, b, c and d.

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Outcomes

0580E2.13A

Understand the idea of a derived function. Use the derivatives of functions of the form ax^n, and simple sums of not more than three of these.

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