Gradients Flashcards

All 7 cards in this deck

What is the formula for the gradient of the line through $(x_1,y_1)$ and $(x_2,y_2)$?
e.g. through $(5,-4)$ and $(7,2)$

$m = \dfrac{y_2-y_1}{x_2-x_1}$
e.g. $\dfrac{2-(-4)}{7-5} = 3$

In the equation $y = mx + c$, what do $m$ and $c$ represent?

$m$ is the gradient and $c$ is the $y$-intercept (where the line crosses the $y$-axis).

What is true of the gradients of two parallel lines?

They are equal: $m_1 = m_2$ (same gradient, different $y$-intercept).

What is the relationship between the gradients of two perpendicular lines?

$m_1 \times m_2 = -1$ — each gradient is the negative reciprocal of the other.
e.g. $\tfrac45$ and $-\tfrac54$

True or false? The gradient of the line $y = 3x + 4$ is $3x$.

False. The gradient is $3$ — a number, with no $x$ in it.

What are the steps to find an unknown coordinate when the gradient is given?
e.g. $P(1,k)$, $Q(9,6)$ and the gradient of $PQ$ is $2$

1. Set up the gradient equation: $\dfrac{6-k}{9-1} = 2$
2. Multiply out: $6 - k = 16$
3. Solve: $k = -10$

True or false? Any two distinct points on a straight line give the same value for the gradient.

True. The gradient of a straight line is constant, so any two points on it can be used in $\frac{y_2-y_1}{x_2-x_1}$.