4.1 Gradients Worksheet

A* Gradients of Lines and Curves 40 Marks

1. Find the equation of the line parallel to the given line which passes through the given point. Give your answer in the form $ay+bx=c$

1. $3x-4y=12$ through (4,-1)
2. $5y-2x=10$ through (-4,2)
3. $3x+5y=12$ through (5,0)
4. $4x+2y=5$ through (1,1)

[8 marks]

2. Find the equation of the line perpendicular to the given line which passes through the given point. Give your answer in the form $ay+bx+c=0$

1. $2x+3y=6$ through (6,1)
2. $3y-4x=5$ through (-4,-3)
3. $5y+3x=8$ through (2,0)
4. $4x-2y=3$ through (1,0)

[8 marks]

3.

The graph shows a plot of a cubic function.

1. Find the values of x for which the gradient is 0
2. Draw a tangent line at x=1 and use this to estimate the gradient

[4 marks]

4.

The graph shows a plot of a quadratic function.

1. Find the value of x for which the gradient is 0
2. Draw a tangent line at x=2 and use this to estimate the gradient

[4 marks]

5. Start by expanding $(x+h)^2$ and subsequently multiplying by $(x+h)$ to find an expression for $(x+h)^5$. Use the result to find the gradient function for $y=x^5$

[5 marks]

6. Find the gradient funtion for $y=x^2+x$ by looking at the limit of the following as $h\rightarrow 0$ \[ \dfrac{(x+h)^2+(x+h)-(x^2+x)}{h}\]

[5 marks]

7. Can you conjecture a result about the gradient function for $y=x^n$? Can you prove this result?

[6 marks]