Sec 3 E-Math · Graphs of functions

Gradient of a curve by tangent questions with answers

Practice questions on gradient of a curve by tangent, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Each diagram shows the curve $y = x^{3} - 2x + 1$ , the point $P$ on it where $x = 0$ , and a straight line. In which diagram is the line the tangent to the curve at $P$ ?

Step 1 of 4

A tangent at P passes through P

Rule out any line that misses

P

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Questions

Question 1Spot the tangent
Each diagram shows the curve $y = x^{3} - 2x + 1$ , the point $P$ on it where $x = -1$ , and a straight line. In which diagram is the line the tangent to the curve at $P$ ?
A
B
C
D
Show the worked solution
1. Step 1: A tangent at P passes through PRule out any line that misses $P$ .
2. Step 2: It touches the curve at P without cutting across itNear $P$ , the curve stays on one side of the line.
3. Step 3: Check each lineDiagram $A$ : the line cuts the curve at two points: it is a chord, not a tangent.Diagram $B$ : the line is the tangent at $P$ .Diagram $C$ : the line touches the curve, but not at $P$ .Diagram $D$ : the line goes through $P$ but crosses the curve there, at the wrong angle.
4. Step 4: AnswerDiagram $B$
Question 2Quadratic curve, points given
The diagram shows the curve $y = x^{2} - 2x$ and the tangent to the curve at $x = 0$ . Use the two points marked on the tangent to find the gradient of the curve at $x = 0$ .
Show the worked solution
1. Step 1: Take the two points on the tangent $A = (-1, 2)$ , $B = (1, -2)$
2. Step 2: The gradient of the curve is the gradient of the tangent: put the two points into the gradient formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ $m = \dfrac{-2 - 2}{1 - (-1)}$ $\phantom{m} = -2$
Question 3Cubic and reciprocal curves
The diagram shows the curve $y = -\dfrac{4}{x}$ and the tangent to the curve at $x = 2$ . Use the two points marked on the tangent to find the gradient of the curve at $x = 2$ .
Show the worked solution
1. Step 1: Take the two points on the tangent $A = (1, -3)$ , $B = (3, -1)$
2. Step 2: The gradient of the curve is the gradient of the tangent: put the two points into the gradient formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ $m = \dfrac{-1 - (-3)}{3 - 1}$ $\phantom{m} = 1$
Question 4Choose your own points
The diagram shows the curve $y = \dfrac{4}{x} + 1$ and the tangent to the curve at $x = 1$ . Choose two points on the tangent and use them to find the gradient of the curve at $x = 1$ .
Show the worked solution
1. Step 1: Pick two points on the tangent, far apart, where it crosses grid lines exactlyFirst point: $(0, 9)$ Second point: $(2, 1)$
2. Step 2: Gradient = change in y ÷ change in x, using your two points $m = -4$
Question 5Real world: speed and rate
The graph shows the height of a plant, $h$ cm, after $t$ weeks. The tangent at $t = 3$ has been drawn. Using the tangent, find the rate of growth, in cm per week, when $t = 3$ .
Show the worked solution
1. Step 1: Take the two points on the tangent $A = (2, 3)$ , $B = (4, 15)$
2. Step 2: The gradient of the curve is the gradient of the tangent: put the two points into the gradient formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ $m = \dfrac{15 - 3}{4 - 2}$ $\phantom{m} = 6$
3. Step 3: The gradient is the rate of growthRate of growth $= 6$ cm per week

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Worked example

Each diagram shows the curve

y = x^{3} - 2x + 1

, the point

P

on it where

x = 0

, and a straight line. In which diagram is the line the tangent to the curve at

P

Step 1 of 4

A tangent at P passes through P

Rule out any line that misses

P

Questions

Question 1Spot the tangent

Each diagram shows the curve $y = x^{3} - 2x + 1$ , the point $P$ on it where $x = -1$ , and a straight line. In which diagram is the line the tangent to the curve at $P$ ?

Show the worked solution

Step 1: A tangent at P passes through PRule out any line that misses $P$ .
Step 2: It touches the curve at P without cutting across itNear $P$ , the curve stays on one side of the line.
Step 3: Check each lineDiagram $A$ : the line cuts the curve at two points: it is a chord, not a tangent.Diagram $B$ : the line is the tangent at $P$ .Diagram $C$ : the line touches the curve, but not at $P$ .Diagram $D$ : the line goes through $P$ but crosses the curve there, at the wrong angle.
Step 4: AnswerDiagram $B$

Question 2Quadratic curve, points given

The diagram shows the curve $y = x^{2} - 2x$ and the tangent to the curve at $x = 0$ . Use the two points marked on the tangent to find the gradient of the curve at $x = 0$ .

Show the worked solution

Step 1: Take the two points on the tangent $A = (-1, 2)$ , $B = (1, -2)$
Step 2: The gradient of the curve is the gradient of the tangent: put the two points into the gradient formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ $m = \dfrac{-2 - 2}{1 - (-1)}$ $\phantom{m} = -2$

Question 3Cubic and reciprocal curves

The diagram shows the curve $y = -\dfrac{4}{x}$ and the tangent to the curve at $x = 2$ . Use the two points marked on the tangent to find the gradient of the curve at $x = 2$ .

Show the worked solution

Step 1: Take the two points on the tangent $A = (1, -3)$ , $B = (3, -1)$
Step 2: The gradient of the curve is the gradient of the tangent: put the two points into the gradient formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ $m = \dfrac{-1 - (-3)}{3 - 1}$ $\phantom{m} = 1$

Question 4Choose your own points

The diagram shows the curve $y = \dfrac{4}{x} + 1$ and the tangent to the curve at $x = 1$ . Choose two points on the tangent and use them to find the gradient of the curve at $x = 1$ .

Show the worked solution

Step 1: Pick two points on the tangent, far apart, where it crosses grid lines exactlyFirst point: $(0, 9)$ Second point: $(2, 1)$
Step 2: Gradient = change in y ÷ change in x, using your two points $m = -4$

Question 5Real world: speed and rate

The graph shows the height of a plant, $h$ cm, after $t$ weeks. The tangent at $t = 3$ has been drawn. Using the tangent, find the rate of growth, in cm per week, when $t = 3$ .

Show the worked solution

Step 1: Take the two points on the tangent $A = (2, 3)$ , $B = (4, 15)$
Step 2: The gradient of the curve is the gradient of the tangent: put the two points into the gradient formula $m = \dfrac{y_2 - y_1}{x_2 - x_1}$ $m = \dfrac{15 - 3}{4 - 2}$ $\phantom{m} = 6$
Step 3: The gradient is the rate of growthRate of growth $= 6$ cm per week