Copy and complete the table of values for the relation y=3x2−5x−7.
x
−3
−2
−1
0
1
2
3
4
y
35
−7
−9
5
Model answer
x
−3
−2
−1
0
1
2
3
4
y
35
15
1
−7
−9
−5
5
21
For example x=−2: 12+10−7=15, and x=4: 48−20−7=21.
(b)
Using scales of 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, draw the graph of y=3x2−5x−7 for −3≤x≤4.
Model answer
Plot every point from the table, then join them with one smooth curve (not straight lines between points). Scale: 2 cm to 1 unit across, 2 cm to 5 units up.
For (c): (i) roots x≈−0.9 and 2.6; (ii) the minimum is about −9.1 (at x≈0.8); (iii) draw the tangent at (2,−5) and measure its slope using two points on it far apart: the gradient is about 7.
(c)
From your graph: (i) find the roots of the equation 3x2−5x−7=0; (ii) estimate the minimum value of y; (iii) calculate the gradient of the curve at the point x=2.
Try it on a graph
The curve and its tangent at x = 2 (gradient 7).
Worked solution (try it first)
(a)
Substitute each x into y=3x2−5x−7.
For example, x=−2 gives 12+10−7=15 and x=4 gives 48−20−7=21.
x
−3
−2
−1
0
1
2
3
4
y
35
15
1
−7
−9
−5
5
21
(b)
With 2 cm to 1 unit on the x-axis and 2 cm to 5 units on the y-axis, plot the eight points and join them with a smooth U-shaped curve.
(c)(i)
The roots are where the curve crosses the x-axis: x≈−0.9 and x≈2.6.
(ii)
The lowest point of the curve is a little right of x=1: the minimum value is y≈−9.1.
(iii)
Draw the tangent at (2,−5).
It passes through about (1,−12) and (3,2), so the rise is 2−(−12)=14 and the run is 3−1=2.