18.6 Estimating the Gradient of a Quadratic Curve Using a Tangent

🟦 Concept

A quadratic graph is a curve, so its gradient changes from point to point.

To estimate the gradient at a particular point, draw a tangent to the curve at that point.

A tangent is a straight line that touches the curve at the point where the gradient is required.

Once the tangent is drawn, its gradient can be calculated in the same way as the gradient of any straight line.

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The tangent line is not drawn properly, this is common problem, you need to practice to make a proper tangent so that it only touches one point on Parabola.

🟩 Formula

The gradient of a straight line is:$$\text{Gradient}=\frac{\text{change in }y}{\text{change in }x}$$

Using two points on the tangent:$$m=\frac{y_2-y_1}{x_2-x_1}$$

where $m$ represents the gradient.

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🟪 Example

Consider the quadratic curve:$$y=x^2$$

Suppose we want to find the gradient of the curve at:$$(2,4)$$

First, draw a tangent that touches the curve at $(2,4)$.

Now choose two convenient points on the tangent.

Suppose these points are:$$(1,0)$$

and:$$(3,8)$$

Use the gradient formula:$$m=\frac{y_2-y_1}{x_2-x_1}$$

Substitute the coordinates:$$m=\frac{8-0}{3-1}$$$$m=\frac{8}{2}$$$$m=4$$

Therefore, the estimated gradient of the quadratic curve at $(2,4)$ is:$$4$$

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🟨 How to Draw the Tangent

Place a ruler so that it touches the curve at the required point and follows the direction of the curve at that point.

Then draw a straight line through the point.

The tangent should not cut sharply across the curve.

After drawing the tangent, choose two clear points on the tangent to calculate its gradient.

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🟧 Understanding the Gradient

If the curve is rising from left to right, the gradient is:$$\text{positive}$$

If the curve is falling from left to right, the gradient is:$$\text{negative}$$

At the turning point of a quadratic curve, the tangent is horizontal.

Therefore:$$\text{Gradient}=0$$

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💡 Exam Tip

When calculating the gradient, choose two points on the tangent that are far apart.

A larger triangle usually gives a more accurate estimate.

The two points you use do not have to lie on the quadratic curve. They must lie on the tangent line.

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⚠️ Common Mistake

Do not calculate the gradient using two random points on the quadratic curve.

The formula:$$m=\frac{y_2-y_1}{x_2-x_1}$$

must be applied to two points on the tangent when estimating the gradient at one point on a curve.

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⭐ Quick Summary

To estimate the gradient of a quadratic curve:

Choose the required point

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Draw a tangent at that point

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Choose two points on the tangent

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Calculate the change in y and change in xx

↓$$m=\frac{y_2-y_1}{x_2-x_1}$$

Remember

Gradient of a curve at a point = gradient of the tangent at that point