13.15 Gradient
Describe the gradient of the lines in the following graphs:
For each of the following intervals between points A and B:
Find the rise.
Find the run.
Find the gradient.
A \left(2, 4\right) and B \left(8, 6\right)
A\left( - 3 , 4\right) and B\left(3, 16\right)
A \left( - 8 , 4\right) and B \left( - 1 , 18\right)
A \left( - 5 , - 2 \right) and B \left(-1, 10\right)
A \left( - 2 , - 1 \right) and B \left(2, - 13 \right)
A \left( - 1 , 2\right) and B \left(1, - 4 \right)
A \left(1, 3\right) and B \left(7, - 3 \right)
A \left( - 3 , - 2 \right) and B \left(5, 6\right)
Explain why a vertical line has an undefined gradient.
State the gradient of any line parallel to the x-axis.
Find the gradient of the following lines passing through points A and B:
Find the gradient of the interval joining A \left( - 9 , 4\right) and B \left( - 3 , - 5 \right).
If we have two points and the slope formula m = \dfrac{y_2 - y_1}{x_2 - x_1}, does it matter which point is \left(x_1, y_1\right) and which point is \left(x_2, y_2\right)?
Consider the line plotted, where A \left(2, 0\right) and B \left(0, 4\right) both lie on the line.
Solve for the gradient of the line.
As x increases, what happens to the value of y?
Find the gradient of the line that passes through the given points:
\left( - 1 , 0\right) and \left(0, 3\right)
\left( - 4 , 7\right) and \left(1, 10\right)
\left(1, - 4 \right) and the origin
\left(2, - 6 \right) and the origin
\left(6, 4\right) and \left(3, 4\right)
\left( - 6 , 5\right) and \left(4, 5\right)
\left( - 2 , - 5 \right) and \left( - 9 , - 12 \right)
\left( - 3 , - 1 \right) and \left( - 5 , 1\right)
Consider the points A \left( - 11 , - 9 \right), B \left( - 5 , 1\right) and C \left( - 2 , 6\right):
Find the gradient of AB.
Find the gradient of BC.
Do the points A, B and C lie in a straight line?
Consider the following points: A \left(26, m - 24\right), B \left( - 1 , m\right) and C \left( - 10 , 9\right).
Find m, given that A, B and C are collinear.
Given the gradient of the line passing through the two points, find the value of the pronumeral:
\left(4, - 3 \right) and \left(1, t\right), gradient = - 2
\left(5, 3\right) and \left(2, t\right), gradient = - 4
\left(5, 3\right) and \left(d, 63\right), gradient = 4
\left(11, c\right) and \left( - 20 , 16\right), gradient = - \dfrac{4}{7}
The line x = 4 intersects the line y = 2 x - 10 at the point Q.
The line x = - 4 intersects the line y = 2 x - 10 at point R.
Find the coordinates of Q.
Find the coordinates of R.
Find the gradient of a line that passes through Q and R.
Consider the line y = 5 x + 2 graphed:
Find the y-value of the point on the line where x = 5.
Find the gradient of the line.
Consider the line y = - 2 x + 8.
Find:
y-intercept
x-intercept
Sketch the line on a number plane.
Hence, find the gradient of the line.
Two lines L_{1} and L_{2} have equations \\y = x - 3 and y = - x + 5 respectively. The lines and their point of intersection have been graphed:
When x = 4, find the y-coordinate of the corresponding point on L_{1}.
When x = 4, find the y-coordinate of the corresponding point on L_{2}.
Find the gradient of the two lines:
L_{1}
L_{2}
Find the product of their gradients.
Consider the quadrilateral ABCD that has been graphed on the number plane:
Find the gradient of following sides:
AB
CD
AD
BC
What type of quadrilateral is ABCD? Explain your answer.
The 4 vertices of square ABCD have been plotted on a number plane.
Find the gradient of side AB.
Find the gradient of side BC.
Find the product of the gradients in parts (a) and (b).
If two lines are perpendicular their gradients multiply to -1. Are sides AB and BC perpendicular?
The points P \left(0, - 2 \right), Q \left( - 2 , 0\right), R \left(0, 4\right) and S \left(2, 2\right) are graphed below:
Find the gradient of RS.
The vertices of \triangle ABC are A \left(9, - 12 \right), B \left(4, 4\right) and C \left( - 8 , - 5 \right). The sides AB and AC have midpoints D and E respectively.
Find the coordinates of points D and E.
Find the gradient of side BC.
Find the gradient of side DE.
Are BC and DE parallel to each other?
A \left( - 4 , - 2 \right), B \left(2, 1\right) and C \left(2, - 4 \right) are the vertices of a triangle.
Name the side of the triangle that is a vertical line.
Find the area of the triangle.
Consider the following ramp:
Find the gradient of this skateboard ramp if it rises 0.9 metres above the ground and runs 1 metre horizontally at the base.
The ramp can only be used as a 'beginner’s ramp' if for every 1 metre horizontal run, it has a rise of at most 0.5 metres. Can it be used as a 'beginner’s ramp'?
A certain ski resort has two ski runs as shown in the diagram:
Find the gradient of Run A. Round your answer to two decimal places.
Find the gradient of ski run B. Round your answer to two decimal places.
Which run is steeper?
A paratrooper falls to the ground along a diagonal line. His fall begins 1157 \text{ m} above the ground, and the line he follows has a gradient of 1.3. That is, he falls 1.3 \text{ m} vertically for every 1 \text{ m} he moves across horizontally.
How far horizontally across the ground does he land from his initial position in the sky?