Lesson 1, Topic 2
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# The Gradient of a Curve

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Recall that the gradient of a straight line (a tangent) is constant at any point on the line. However, the gradient of a curve changes from point to point along the curve.

Consider the curve of y = x2 + 2x – 4 above:

Note that the gradient of the points P and S is the same as the gradient of PQ and ST gradient, respectively.Â

Gradient at P = $$\frac{y_2 \: – \: y_1}{x_2 \: – \: x_1} \\ = \frac{3\: – \: (-1)}{2\: – \: 1} \\ = \frac {4}{1} \\ = \scriptsize 4$$

Also Gradient at S = $$\frac{y_4 \: – \: y_3}{x_4 \: – \: x_3} \\ = \frac{-2\: – \: (-4)}{-3\: – \: (-2)} \\ = \frac {2}{-1} \\ = \scriptsize -1$$

It is important to note that the gradient at point P is positive because the tangent $$\scriptsize \overline {PQ}$$ slopes upward from left to right. Also, the gradient at point S is negative because the tangent $$\scriptsize \overline {ST}$$ slopes downwards from left to right.

Given y = ax2 + bx + c

The graphs below show that the gradient at the turning points is zero because the tangents at points P and Q (turning points) are parallel to the x-axis.

Point P and Q are called turning points. Point P is the maximum point while point Q is the minimum point. Note that the line of symmetry in each case passes through the turning point.

### Example 1

(a) Copy and complete the table below

For the value of the relation y = 11 – 2x – 2x2 ………for  -4 â‰¤ x â‰¤ 3.

(b) Using a scale of 2cm to 1 unit on the x-axis, draw the graph of y = 11 – 2x -2x2

(c) Use your graph to find:

(i) The roots of the equation 11 – 2x – 2x2 = 0
(ii) The value of x for which 3 – 2x – 2x2 = 0
(iii) The gradient of the curve at x = 1 Â  Â  Â  Â  Â  Â  Â  (SSCE)

Solution: (a)

(b) Diagram of Quadratic Function Graph

Scale: 1 unit â‰¡2cm on x-axis

Â Â Â Â Â Â Â Â Â Â Â Â 5 units â‰¡ 2cm on y-axis

(c) (i) roots: x = -2.9 or      x = 1.9

(ii) 3 – 2x – 2x2 = 0

y + (3 – 2x – 2x2) = 11 – 2x – 2x2

y = 11 – 2x – 2x2 – 3 + 2x + 2x2

y = 8

values: x = 0.8 or -1.8 â‡’ y = 8

(iii) The gradient of the curve at x = 1

To find the gradient at x = 1, we will need to draw a tangent at this point as shown in the graph below.

Gradient at (x = 1) = $$\frac{y_2 \: – \: y_1}{x_2 \: – \: x_1} \\ = \frac{3.4\: – \: 7}{1.6\: – \: 1} \\ = \frac {3.6}{0.6} \\ = \scriptsize 6$$

### Example 2

(a) Copy and complete the following table for y = 2x2 -7x -3

(b) Using 2cm to 1unit on the x-axis and 2cm to 5units on the y-axis,
draw the graph of y = 2x2 – 7x – 3………..for -2 â‰¤ x â‰¤5.

(i) Minimum value of y
(ii) Gradient of the curve at x = 3 Â  Â  Â  Â  Â  Â  Â  Â  (SSCE)

Solution:

Diagram of a Quadratic Function Graph

(i) ymin = -9.1

(ii) To find the gradient at x = 3, we will need to draw a tangent at this point as shown in the graph below.

Gradient = $$\frac{y_2 \: – \: y_1}{x_2 \: – \: x_1} \\ = \frac{-3 \: – \: (-6)}{3.7 \: – \: 3}$$

Gradient = $$\frac{3}{0.7} \scriptsize = 4.29$$

â‡’ Gradient of curve (at x = 3) = 4.

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