Estimating the Gradient of a Curve Answers
1. Find the gradient of each line segment.
a.
3
b.
y
3
c.
y
2
2
2
1
1
1
x
-3
-2
-1
3
0
1
2
3
x
-3
-2
-1
0
1
2
3
x
-3
-2
-1
0
-1
-1
-1
-2
-2
-2
-3
-3
-3
gradient = 2
gradient = -4
gradient = 5
d.
3
e.
y
3
2
-1
0
2
3
4
y
1
x
-2
1
2
1
-3
y
1
2
3
x
-3
-2
-1
0
-1
-1
-2
-2
-3
-3
gradient = 1
gradient = - 2
2
1
2
3
3
2. Find the gradient between each pair of coordinates.
a. (3, 5) and (2, 1)
gradient =
5–1
=4
3–2
b. (3, -2) and (0, 0)
gradient = -2 – 0 = - 2
3–0
3
c. (-1, 2) and (7, 5)
gradient = 2 – 5 = 3
-1 – 7
8
d. (7, 5) and (3, -2)
(-2) 7
=
7–3
4
gradient = 5 –
e. (3, 0) and (-2, 4)
gradient = 0 – 4 = - 4
3 – (-2)
5
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Estimating the Gradient of a Curve Answers
3. Match each of the graphs below with the correct interpretation of its gradient. If there is not
a matching pair, write one.
a distance-time graph
the interest rate
a velocity-time graph
the speed
a speed-time graph
the rate of change of the water
a graph of the amount of money owed
the acceleration
a graph of the amount of water in a tub
the acceleration
4. For each of the graphs below, estimate the gradient at the given value of x.
The gradients given are estimates. Answers should be considered correct if they are within
0.2 of the given answer.
a.
at x = 2
5
b.
at x = 1
y
3
4
y
2
3
1
2
-1
1
0
1
2
3
4
x
5
x
3
2
gradient = 1.6
at x = 3
10
1
-1
gradient = -0.2
c.
0
d.
y
at x = 4
3
y
8
2
6
4
1
2
0
2
gradient = 1
4
6
x
8
0
1
2
gradient = 0.25
3
4
5
x
6
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Estimating the Gradient of a Curve Answers
5. The graph below is a distance-time graph for a sprint runner. Find the speed of the runner
after 5 seconds.
20
distance (m)
15
10
5
0
5
10
time (s)
15
The gradient at 5 is
1.5, therefore the
speed of the runner is
approximately 1.5m/s.
6. The graph below is a velocity-time graph for a new racing car.
60
a. The manufacturer claims the car can
go from 0m/s to 45m/s in 3 seconds.
Comment on the manufacturer’s claim.
50
The speed at 3 seconds is less than
45m/s so, in this case, the car has
not done 0m/s to 45m/s in 3 seconds.
velocity (m/s)
40
30
b. Calculate the acceleration of the car at
4 seconds.
20
The gradient at 4 seconds is
approximately 4, therefore the
2
acceleration is approximately 4m/s
10
0
2
4
time (s)
6
8
10
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Estimating the Gradient of a Curve Answers
7. The speed-time graph below shows a car braking.
a. Interpret the y-intercept of the
graph.
15
The y-intercept represents the
initial speed of the car, 15m/s.
b. Estimate the deceleration of the
graph after 5 seconds.
speed (m/s)
10
The gradient at 5 seconds is
approximately -1, therefore the
2
deceleration is 1m/s
5
0
2
4
6
time (s)
8
10
12
8. The graphs below represent two rowing squads in a race. Graph 1 is a distance-time graph
for squad 1. Graph 2 is a speed-time graph for squad 2.
Graph 1
700
a. Describe Graph 2, referencing
the speed and acceleration of the
rowing squad during the race.
distance (m)
600
500
400
300
200
100
0
10
20
30
40
50
time (s)
60
70
80
Graph 2
14
b. After 10 seconds, which squad
is travelling faster? Justify your
answer.
speed (m/s)
12
10
8
Squad 1: the gradient at 10
seconds is roughly 10, so their
speed is 10m/s
6
4
2
0
The squad accelerate quickly
in the first 10 seconds, then
continue to accelerate at a slower
rate until 50 seconds. They then
travel at a constant speed of
12m/s until 65 seconds where
they accelerate again for the last
5 seconds of the race.
Squad 2: their speed is 8.5m/s
10
20
30
40
50
time (s)
60
70
80
Squad 1 is travelling faster.
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Estimating the Gradient of a Curve Answers
Challenge
Below is the graph of y = x2. Complete the table with the gradient of the graph for each value of x.
What do you notice?
10
y
9
8
7
6
5
4
3
2
1
x
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
0.5
1
1.5
2
2.5
3
Gradient
1
2
3
4
5
6
Each gradient is approximately double the value for x.
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