What tells you whether a gradient should be positive or negative?

The direction of the line decides it, with uphill lines running from bottom left to top right having a positive gradient. A downhill line running from top left to bottom right has a negative gradient.

National 5MathsSQAFlashcardsAlgebraic SkillsEquation of a Straight Line

Equation of a Straight Line (SQA National 5 Maths): Flashcards

Exam code: X847 75

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FrontGradient of a Line
Define the gradient of a straight line.

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Cards in this collection (15)

Define the gradient of a straight line.
The gradient measures how steep a line is, giving the change in $y$ for every $1$ unit moved to the right.
A gradient of $3$ means going up $3$ , and a gradient of $- 4$ means going down $4$ .
Complete the gradient formula by filling in the two missing expressions.
$gradient between (x_{1}, y_{1}) and (x_{2}, y_{2}) = \frac{______}{______}$
The completed formula is:
$gradient = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
The $y$ values go on top and the $x$ values underneath, and both differences must be taken in the same order.
How do you find a gradient from a line drawn on a grid?
Pick two points on the line and draw a right-angled triangle, then work out $\frac{rise}{run}$ .
The rise is the vertical length and the run the horizontal one, so a rise of $3$ over a run of $1$ gives a gradient of $3$ .
True or False?
A gradient of $- 5$ is steeper than a gradient of $- 4$ .
True.
Steepness depends on the size of the gradient rather than on its sign, and $5$ is bigger than $4$ .
Both lines slope the same way, so the one with gradient $- 5$ is the steeper of the two.
What tells you whether a gradient should be positive or negative?
The direction of the line decides it, with uphill lines running from bottom left to top right having a positive gradient.
A downhill line running from top left to bottom right has a negative gradient.
A line joins $(- 1, 6)$ and $(3, - 2)$ so what is its gradient?
Substituting into the formula gives $\frac{- 2 - 6}{3 - (- 1)} = \frac{- 8}{4}$ , so the gradient is $- 2$ .
Subtracting a negative $x$ value is the step most easily got wrong, since $3 - (- 1)$ is $4$ and not $2$ .
How do you draw a line with a gradient of $- 2$ from a given point?
Write the gradient as a fraction, $- \frac{2}{1}$ , so that the rise is $2$ and the run is $1$ .
Because the gradient is negative, move $1$ unit to the right and $2$ units down from the point, then join up.
In the equation $y = m x + c$ what do $m$ and $c$ stand for?
$m$ is the gradient and $c$ is the y-intercept, the value where the line cuts the $y$ axis.
So $y = 3 - 4 x$ has gradient $- 4$ and a $y$ axis crossing at $3$ .
To use $y - b = m (x - a)$ what two pieces of information do you need?
You need the gradient $m$ and the coordinates of one point $(a, b)$ that the line passes through.
For gradient $3$ through $(2, 5)$ the equation is $y - 5 = 3 (x - 2)$ .
A line has gradient $- 2$ and passes through the point $(8, 6)$ so complete its equation.
$y -______= - 2 (x -______)$
The completed equation is:
$y - 6 = - 2 (x - 8)$
The $y$ coordinate goes with the $y$ and the $x$ coordinate with the $x$ , so a single point supplies both numbers.
How is $y - 5 = 3 (x - 2)$ rearranged into $y = m x + c$ form?
Expand the bracket and then make $y$ the subject, so $y - 5 = 3 x - 6$ becomes $y = 3 x - 6 + 5$ .
Simplifying the numbers gives $y = 3 x - 1$ .
How can $y = m x + c$ itself be used to find the equation of a line?
Substitute the gradient and the coordinates of a known point into $y = m x + c$ , then solve for $c$ .
With $m = - 2$ and the point $(2, 18)$ that gives $18 = - 2 (2) + c$ , so $c = 22$ and the line is $y = - 2 x + 22$ .
True or False?
A vertical line has an equation of the form $y = c$ .
False.
A horizontal line has the equation $y = c$ , where $c$ is the value at which it crosses the $y$ axis.
A vertical line has the equation $x = k$ , such as $x = - 2$ .
When finding the equation of a line through two points does the choice of point matter?
Either point may be used, because both lie on the same line and both lead to the same final equation.
For the line through $(2, 18)$ and $(8, 6)$ with gradient $- 2$ , both choices simplify to $y = - 2 x + 22$ .
What can you do when a graph does not show where the line crosses the $y$ axis?
Find the gradient from the graph, take the coordinates of any point the line passes through, and use one of the two methods to build the equation.
The crossing point then comes out of the working rather than being read off.

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